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THE 



STUDENT'S HANDBOOK 



SYNOPTICAL AND EXPLANATORY 



ME J. S. MILL'S SYSTEM OF LOGIC 



BY 

KEV. A. H. KILLICK, M.A. 

FELLOW OF THE UNIVERSITY OF DURHAM 



LONDON 
LONGMANS, GREEN, HEADER, AND DYER 

1870 






PRINTED BY BALLANTYNE AND COMPANY 

EDINBURGH AND LONDON 

Exchang* 
Univ. of 
f£B 3 IflSfc 



PREFACE. 



The design of this Handbook is to facilitate as much 
as possible the study of Inductive Logic, — particu- 
larly as represented in Mr J. S. Mill's volumes on 
the subject. It is therefore, in the main, an epitome 
of that work, the arguments being condensed and 
summarised, the necessary explanations being given 
wherever it seemed likely that a student would feel 
any difficulty, and the whole being so arranged that 
the connexion and relative importance of the different 
topics discussed may be recognised at a glance. The 
single aim of the author has been to render the work 
what its name imports — a Handbook to aid the care- 
ful study of the original, by furnishing the reader, 
chapter by chapter, with such a coup d'ceil of the 
subject as may best prepare him for a thorough and 
intelligent comprehension of Mr Mill's system. 

Those whose logical reading has been confined to 



IV PREFACE. 

Whately or the common manuals of the science, may 
perhaps be not a little perplexed, on directing their 
attention for the first time to the study of Mill, by 
the total difference in the manner in which the entire 
subject appears to be treated. Many topics which 
are entirely omitted, or very slightly treated, in the 
most popular logics, or if mentioned, are mentioned 
only to be expressly excluded from the domain of the 
science, are elaborately discussed by Mr Mill ; who, 
on the other hand, passes over, with scarcely any 
notice, many subjects which occupy a large space in 
the treatises of most other logical writers. Some of 
these differences are merely such as would occur 
between any two independent thinkers discussing the 
same subjects ; some are connected with differences 
of opinion on certain metaphysical points, which, 
though themselves no part of logical science, neces- 
sarily modify the views which are taken of logical 
questions ; but in general they depend upon a more 
fundamental cause, a due consideration of which will 
not only often explain the apparent or real discre- 
pancies between different writers on logic, to which 
we have alluded, but will often also throw light on 
some of the most perplexing and hopelessly entangled 
questions with which Logical Science is concerned. 



PREFACE. V 

The explanation referred to will be found to a great 
extent to be involved in the distinction between what 
may be termed respectively "Objective" and " Sub- 
jective Inference" — a distinction of great importance, 
and one which it is essential that the student should 
thoroughly comprehend. 

In " Objective Inference" the fact stated in the 
conclusion is a bona fide new truth, a distinct fact, 
and not merely part of the same fact or facts stated 
in the premisses. Thus, if we find that half-a-dozen 
pieces of loadstone possess each the property ot 
attracting iron, and hence infer that a seventh piece 
which we have not tried will also manifest the same 
property, it is perfectly clear that this last fact is 
something new, and by no means included in the 
previous facts (that the six loadstones attract iron) 
which form the premisses of our conclusion. In such 
a case, as in all cases of Objective Inference, the con- 
clusion follows in virtue of a law of External Nature 
(hence the designation " Objective "), and not by a mere 
law of mind ; for no contradiction or impossibility is 
involved in thinking that the first six objects possess 
the property in question, while a seventh does not ; 
and whether it does or does not will evidently be a 

mere question of physical law. A consequence of 

b 



VI PREFACE. 

this is, that such inferences cannot be expected in 
symbols in such a way that the conclusiveness of the 
argument is evident from the mere form >-, — i.e., what- 
ever meaning we choose to assign to the symbols. 
" Objective Inference " is the "Induction" of Mill; 
with other logical writers it is usually spoken of as 
" Material Induction/' and is not only contrasted 
with what they call " true logical Induction " (which 
we shall find to be the same with Mr Mill's " Mere 
Verbal Transformation"), but is by them expressly 
excluded as a subject whose consideration ought to 
form no part of Logical Science. 

"Subjective Inference" on the other hand, affords 
a contrast in all these respects. It is, in short, an 
explicit statement of a fact drawn from premisses in 
which it was in reality implied, so that the mind, 
being in possession of the premisses, can, by a mere 
comparison of their expression in words, evolve the 
conclusion, — the fact stated in that conclusion being 
really included in — being, in truth, part of — the fact 
or facts stated in the premisses. 

Thus, if from the proposition "All men are 
mortal" we draw the conclusion that some particular 
individual, A, will also at some time or oilier die, it 
is clear that this last is really involved in the previous 



PEEFACE. Vll 

statement, and we could not believe the former and 
disbelieve the other without violating a law of the 
mind itself — without, in fact, being guilty of a con- 
tradiction. Hence this form of Inference may be 
expressed in symbols, in such a way that the infer- 
ence may be seen to follow from the mere form of the 
expression. Thus, putting A for "men," B for 
" mortal " (beings), C for Ca3sar, we have — 

All A is B 

C isA 
therefore C is B. 

Whatever A, B, C may stand for, if we assent to the 
premisses in such a case as this, we cannot refuse 
our belief to the conclusion without a contradiction. 
The terms "formal" and " subjective" inference 
are, in fact, convertible. The " Syllogism " and the 
so-called " Immediate Inferences " are the principal 
forms which subjective inference assumes. 

If now the distinction which has been pointed 
out be understood, its application is easy. Logic 
is defined to be the " Science of Inference ; " the 
majority of logicians, and particularly Whately, 
Mansel, and Sir William Hamilton, limit its pro- 
vince exclusively to the consideration of subjective 



Vlll PKEFACE. 

inference — in other words, to the Syllogism and 
certain subordinate processes ; objective inference 
they refuse to recognise as coming within the 
domain of the science at all, and relegate it to the 
limbo of the " extra-logical." Whately over and 
over again asserts that all reasoning is Syllogism, 
and that to the consideration of this the duty of the 
logician is strictly confined ; that objective inference 
— the process by which we arrive at new truths — is 
altogether foreign to the subject, and that it is im- 
possible to lay down rules for it. 

Mr Mill, on the other hand, takes a view as directly 
opposed to this as possible ; for he refuses to recog- 
nise what we have above distinguished as subjective 
inference — as being a real inference at all in the 
proper sense of that word — but views the conclusion 
in a Syllogism as being rather an interpretation of, 
than a deduction from, the so-called premisses. 

Further, he regards objective inference, or true 
induction, as not only a part, but by far the most 
important part of the province of Logic, and the 
greater portion of his work is occupied with the con- 
sideration of this process, — its laws, its methods, and 
the genera] conditions of its validity — almost to the 
exclusion of those different forms of formal inference 



PREFACE. IX 

which most other logicians put forward as the only 
possible formula of reasoning. 

Now it is scarcely to be wondered at that authors 
who differ so fundamentally in their views of this 
subject, should differ widely in their mode of treat- 
ing it. The truth seems to be this — Logic is un- 
doubtedly the Science of Inference, and not merely 
of one, but of every form in which that mental opera- 
tion can be presented to us. Such is the principle 
upon which Mill proceeds — that Induction is pro- 
perly within its province as well as Syllogism. 
Whether we decide to apply the term inference to 
the latter or not, — or whether we are content to 
follow Mr Mill in regarding the Syllogism as simply 
a formal mode of interpreting and applying general 
propositions — in any case, its consideration forms an 
important division of the general Science of Logic. 
It will thus be evident that the great apparent dis- 
crepancies between authors like Whately and Hamil- 
ton on the one hand, and Mill on the other, are in 
reality to a great extent not differences at all ; they 
are treating of different branches of the total Science 
of Inference, and apart from inconsistencies neces- 
sarily connected with the different metaphysical 
tenets of the various schools, the general body of 



X PREFACE. 

their logical doctrines may be combined into one 
system, each having its distinct function in the 
generation of scientific belief. A complete treatise 
on Logic would necessarily incorporate material from 
them all, — their systems are not, as wholes, incon- 
sistent, but complementary. 

It only remains to add a word of explanation in 
reference to the arrangement of the present work. 
Each chapter will be found subdivided into sections, 
marked by Eoman numerals in the margin ; in gene- 
ral each such section is limited exclusively to the 
consideration of a single topic. In Book V., on the 
Fallacies, Whately's divisions have been introduced, 
each under its proper heading, in the exhaustive 
classification of Mill ; and those passages throughout 
the work which have not the authority of a direct 
statement in Mr Mill's volumes, are marked by in- 
closure in dark, square brackets. 



CONTENTS. 



PAGE 

Introduction, ...... 1 

BOOK I. 

Names and Propositions, . . . . 6 

BOOK II. 

Inference or Reasoning in General, . , . 55 

BOOK III. 

Induction, ....... 88 

BOOK IV. 
Operations Subsidiary to Induction, . . . 205 

BOOK Y. 

Fallacies, . . . . . . .234 

BOOK VI. 
Logic of Moral Sciences, . • . . 251 

Appendix, ....... 259 



Additions and Errata. 



Preface, page ix. for formula read formulae. 

Page 12, line 3 from bottom, insert (see Appendix). 



91, 
93, 
93, 
97, 
99, 
254, 
256, 



8 „ for theory read theorem. 

16 ,, for motion read notion. 

19, for sense read sum. 
21, insert (see p. 67). 

2, insert (see pp. 177 and 181). 

6 from bottom, for Blatham read Bentham. 
23, for belief read beliefs. 



INTRODUCTION. 



DEFINITION AND SCOPE OE LOGIC. 

Mill, after remarking that the Definition of any 
progressive Science must necessarily be provisional, 
gives two Definitions of Logic, which he criticises, 
before enunciating his own. 

1st Definition. — "Logic is the Science and 
Art of Keasoning." 

[Whately added the term "Science" to this definition, and 
properly so. By "Science'''' is here meant the analysis of 
the mental processes which take place whenever we reason ; 
and by " Art," the rules for properly conducting the pro- 
cess founded upon that analysis.] 

{Meaning of term "Reasoning." In its restricted sense, it is 
equivalent to Syllogizing or Ratiocination ; in a more ex- 
tended and proper sense, it is simply to infer any assertion 
from assertions previously admitted.] 

Criticism of this Definition. — This definition of 
Logic is too narrow, — the theory of argumentation 
merely does not include all that is properly within 
the scope of Logic. Precision of Language (Theory 

id. 



2 INTRODUCTION. 

of Names and Definitions), and Accuracy of Classi- 
fication are almost always reckoned amongst the 
objects of that science ; for we find — 

(a.) The view of professed Logicians is more extensive. 
Most authors, dividing Logic into three parts, treat 
in the first two of Names or Notions and Proposi- 
tions (under one or other of which heads they 
include Definition, Division, &c), and in the third 
part only do they discuss Reasoning. 

(6.) The popular view of Logic, also, is more extensive. In 
common discourse we hear as often of "a logical 
arrangement," or of expressions " logically defined," 
as of conclusions logically deduced from premisses. 

2d Definition. — " Logic is the Science which 
treats of the operations of the Understanding 
in the pursuit of Truth." 

Criticism. — This definition is too wide, because — 

(a.) Logic has nothing to do with one class of Truths, — 
those of Intuition. 

[Truths are known to us in two ways : — 

1. By our immediate consciousness, i.e., by Intuition. 

2. By Inference. 

The truths known to us by Intuition are such as are involved 
in our Feelings, bodily or mental, — that I have such and 
such a feeling ; that I am experiencing such and such an 
emotion or mental state ; that I am conscious, for instance, 
of ;i sensation of blue, or that I feci vexed or hungry, are 
truths of this kind. Such truths are, of necessity, the 
original premisses from which all others are ultimately 

inferred. 

With these original data of our knowledge, Logic has no 
direct concern. Whatever we know in this way,— as, for 



INTRODUCTION. 3 

instance, that I feel hot, or cold, — is known to me beyond 
the possibility of doubt. No Science is required for the 
purpose of testing the validity of our belief in such truths 
as these ; no Art can possibly render our knowledge of 
them more certain. 
Rapid and unconscious inferences must not, however, be con- 
founded with true intuitions.— --"We may really infer what 
we fancy that we actually see or feel ; thus a certain com- 
bination of sensations of form and colour, imprinted as a 
picture on the retina, have always been infallible marks 
of the presence of an external object, as my father; and 
hence, whenever I experience such a combination of sensa- 
tions, I infer his presence, though I may do so wrongly, 
as in delirium or spectral illusion.] 

(b.) It would introduce into Logic many questions with 
which it has no direct concern, many metaphysical 
inquiries especially. 

[Such inquiries as : — 
"What is the nature of Perception, Memory, Belief, or Judg- 
ment ? Are God and Duty realities, the existence of which 
is manifest a priori ? and so forth, — are all connected with 
" the pursuit of truth," but are foreign to Logic] 

Province of Logic and its R elation to Knowledge 
in general. — The province of Logic, then, must be 
restricted to that portion of our knowledge which 
consists of Inferences from data. Every proposition, 
therefore, which is believed as being an inference 
from something else, comes within the scope of the 
principles and tests furnished by Logic. 

The function of Logic as regards that part of our 
Knowledge of which it does take cognizance is* to 
furnish tests whereby we may judge whether the 
inference, by which the knowledge is arrived at, is 
valid or not. The science, or particular subject 



4 INTRODUCTION. 

matter, furnishes the evidence itself. Logic supplies 
the principles or rules for the estimation of the 
(worth of it as) evidence. 

[The Logic of Science the same as the Logic of common life. To 
draw inferences has been said to be the great business of 
human life. — By far the greater portion of our knowledge 
is indubitably matter of inference ; so that not only Science, 
but the great bulk of human beliefs generally, are amen- 
able to logical tests. The business of the physician, gene- 
ral, magistrate, &c, is chiefly to draw inferences from 
data, and to act according to the conclusion therefrom.] 

[Utility of Logic. — "If a Science of Logic exist, that Science 
must be useful ; if there be rules to which every mind 
conforms when it infers rightly, it seems self-evident that 
a person is more likely to act in accordance with those 
rules, if he know them, than if he be ignorant of them."] 

Mill's Definition. — " Logic is the Science of 
the operations of the Understanding, which 
are concerned in, or are subservient to, the 
Estimation of Evidence." 

[In other places Mill gives the following verbally-different forms 
of this Definition : — 
(1.) "Logic is the Science of the investigation of Truth by 

means of Evidence." 
(2.) "Logic is the Science of Inference."] 

Its main subject, then, is Inference ; amongst the 
subsidiary we may notice : — 

(1.) The Theory and Uses of Names and Propositions — 
for language is an important instrument of thought. 

(2.) Definition omd Classification (of which " Division" is 
8 form), wlii.li servo (1.) to preserve our evidence 
and our conclusions in a permanent and convenient 



INTRODUCTION. 5 

form ; and (2.) to marshal our facts in a clear 
order. 

Analysis of Instruments necessary. — The Analysis of the In- 
struments (Names and Propositions) we employ in the 
investigation of Truth is part of the Analysis of the inves- 
tigation itself ; since no art is complete unless another art, 
that of constructing and adjusting the necessary tools, is 
embodied in it.] 

[How far such Analysis must he carried.- — The Analysis of the 
process of Inference, and of the processes thereto subor- 
dinate, need only, for the purposes of Logic, be carried far 
enough to enable us to discriminate between a correct and 
incorrect performance of those processes.] 



BOOK I. 



NAMES AND PROPOSITIONS. 



CHAPTEE I. 

Introductory. 

[The remarks on language in this chapter are, par excellence, 
applicable to general names (whether consisting of one or 
more words), the parts of language with which, and the 
copula, Logic is chiefly concerned.] 

I. The necessity, in Logic, of commencing 
ivith an Analysis of Language. 

" Logic is concerned with a portion of the art of 
thinking. Language is the principal help and in- 
strument of thought, and any imperfection in this 
instrument is confessedly especially liable to confuse 
and impede the main process, and to destroy all 
confidence in the result." 

Answer as far as Reasoning is concerned. — Since 
inference is a process usually carried on by means 



NAMES. 7 

of words, and in complicated cases can be carried on 
in no other way, a thorough insight into the import 
of language is evidently an essential preliminary to 
a correct performance of that process. 

II. Theory of Names especially, why a 
necessary part of Logic. 

Since Logic is the science of proof, or that by 
which we test the validity of the inference of a pro- 
position from propositions already admitted, — it is 
evident that a clear view of the Import of Proposi- 
tions is essential. To this an analysis of Names — 
the chief elements of a proposition — is an indispens- 
able preliminary. 

III. Why Names must he studied before 
Things. 

If one should commence with the study (i.e., ex- 
amination and classification) of Things, without 
using the aid of established Names, 

(1.) No varieties of things would of course be 
included but those personally recognised 
by the individual observer ; and this is to 
discard the results of the labour of all 
preceding observers ; and 

(2.) Even after his personal examination of 
things, it will remain necessary to examine 



5 NAMES. 

names, to be sure that nothing is omitted 
which ought to be included. 
(3.) But by beginning with the classifications 
recognised in common language, and thus 
using names as a clue to the things, we 
bring before us at once all the differences 
and resemblances amongst objects which 
have been recognised, not by a single in- 
quirer merely, but by the collective intelli- 
gence of mankind. 
■t 

IV. Definition and Initial Analysis of Pro- 
positions. 

A Proposition is a sentence in which something 
is affirmed or denied of something. 

It is formed by putting together two names ; 

and consists of the two names — the subject 

and predicate, connected by the copula. 
The Subject is the name denoting that of which 

something is affirmed or denied. 
The Predicate is the name denoting what is 

affirmed or denied of subject. 
The Copula is the sign denoting that there is 

an affirmation or denial. 



NAMES. 



CHAPTEK II. 

Names. 

" A Name is a word or set of words, serv- 
ing the double purpose of — (1.) A mark to 
recall to ourselves the likeness of a former 
thought ; and (2.) Of a sign to make it 
known unto others." 

I. Are Names the names of Things or of 
our Ideas of Things f 

Names are properly the names of things. 

Names are names of our ideas of things in this 
sense only, — that the idea alone, and not the thing 
itself, is recalled by the name, or imparted to the 
hearer. 

But names are not only intended to make the 
hearer conceive what we conceive, but also to inform 
him what we believe ; and therefore it seems proper 
to consider a name as the name of that which we 
intend to be understood by it on the particular occa- 
sion when we make use of it, — that, in short, con- 
cerning which we intend to give information, namely, 



10 NAMES. 

the things themselves. Thus, if I say, " The sun is 
the cause of day," I do not mean that my idea of 
sun causes in me the idea of day, but that the ex- 
ternal thing "sun" is the cause of the external 
phenomenon "day." 

[Or we may sum up thus : — 
Names are names of things ; but an idea of the thing denoted 
by a name is a necessary condition for, or preliminary to, 
the intelligible use of the name. I must have an idea of 
"house," and an idea of "fire," before I can understand 
what is meant by " a house is on fire ; " but the names do 
not refer to those ideas, but to the external objects or 
phenomena themselves.] 

II. Categorematic and Syncategorematic 
Words, and many-worded Names. 

A Categorematic Word is one which can be used 
alone, either as the predicate or subject of a proposi- 
tion. 

A Syncategorematic Word is one which can only 
form a part of a name ; as prepositions, adverbs, &c. 

Many-worded Karnes are such as consist of more 
than one word. 

In Logic a name may consist of any number of 
words, thus, " John Nokes, wlio was mayor of the 
toiun, died yesterday ; " here the first eight words 
form but a single name, the subject of the proposi- 
tion. 

[A ByneaUfforematic loord may be used as the term of a proposi- 
tion by the " su/>/>"sit '<<> mati rial is" i.e. t the speaking of a 
word itself as a thing, —as : "Truly is an English word ;" 
"Heavy is an adjective."] 



NAMES. 11 

III. Distinctions in Names. — Mill discusses 
the following : — 

1. A Singular or Individual Name is such as can 
only be affirmed in the same sense of one thing. 

2. A General Name is a name which can be 
affirmed in the same sense of each of an indefinite 
number of things. 

3. A Collective Name is one which cannot be pre- 
dicated of each individual of a multitude, but only of 
all taken together. 

^Collective and General Names must not be confused ; 
a collective name is really a peculiar form of indi- 
vidual name, the individual being made up of an 
aggregation of precisely similar units. Thus, "A 
regiment" is an individual or aggregate made up 
by repetition of the unit "a soldier;" "a multi- 
tude " of the unit " a human being." But " regi- 
ment," " multitude," are general names, applicable 
to an indefinite number of individual regiments or 
multitudes. J 

4. An Abstract Nome is a name of an attribute. 

5. A Concrete Name is a name of a thing, — that 
is of the thing or things which possess the attribute 
denoted by the corresponding abstract name. (Thus, 
abs. "Fault" cone. " Faulty things;" abs. "Hu- 
manity" cone. " Human Beings" or Men?) 

[Notice that abstract names are either : — 

1. Of single and definite attributes — visibility, hard- 

ness. 

2. Of a group of attributes — humanity. 



12 NAMES. 

3. Of attributes which have attributes, thus, " fault" 
is the name of some attribute which has the 
quality — " causing inconvenience."'] 

6. A Non-connotative Name denotes a subject 
only, or an attribute only, 

7. A Connotative Name denotes a subject (or 
subjects), and implies or involves an attribute (or 
attributes). 

[Thus the name " man " means or connotes certain attri- 
butes — animality, rationality, upright form, &c. 
Other logicians would say the idea of "man" in- 
cludes or " comprehends " the idea of animality, &c. 
The best mode of determining whether a name con- 
notes a given attribute is to ask, Whether, if that 
attribute were removed, the name would still be 
applied to the subjects'? Does "man" connote 
mortality ? The test is, should we apply the name 
" man" to beings exactly like men in other respects, 
but not mortal ?] 
Connotative Names are : — 

(1.) All concrete general names. 
(2.) Abstract names, groups 2 and 3. 
(3.) Certain singular or individual names, — that is, 
names not per se singular, but determined 
somehow to a single individual ; in fact, names 
which describe an individual. Thus the " only 
son of Jones," the " present premier," &c. 
Non-connotative Names are : — 
(1.) Proper names. 
(2.) Names of attributes which have no attributes.] 

8. Positive Name — the simple name of anything, 
as "man" " tree'* 



NAMES. 13 

9. Negative Name — the negation or contradictory 
of a positive name, "not man" "not tree" (includes 
everything else except what is denoted by positive 
name). 

10. Privative Name — equivalent to a positive and 
negative name together, being the name of some- 
thing which might be expected to have a certain 
attribute, but has it not. Thus " blind " = might be 
expected to see, but does not. 

Connotation of these Names. — The negative connotes the 
absence of the attributes connoted by corresponding 
positive ; the privative connote (1.) the absence of 
certain attributes, (2.) the presence of others from 
which the presence also of the former might be 
expected. 

11. Relative Names. — A name is said to be rela- 
tive when over and above the object it denotes, it 
implies in its signification the existence of another 
object, also deriving its denomination from the same 
fact or series of facts which is the ground of the first 
name. 

Thus take any pair of relative names, as parent, offspring. 
Now it is clear that we cannot speak of a parent 
without implying existence of offspring, and vice 
versa. When we call A the parent of B, we under- 
stand that a certain series of events or phenomena 
have happened, in which A and B are both involved. 
This series of facts is implied whenever we speak 
either of parent or offspring; they are the facts con- 
stituting the relation or forming its "fundamentum" 
though the position or attitude in which a parent 



14 NAMES. 

stands to those facts is different from the attitude 
of offspring to them. 

The characteristic property, then, of relative names, is 
that they are always given in pairs ; every relative 
name (as father, son ; like, unlike ; equal, longer), 
which is predicated of an object, supposes another 
object or objects, of which we may predicate either 
the same name, (as " consort "), or another relative 
name, said to be the " correlative " of the former 
(as offspring and parent). 

The Connotation of a relative name, then, consists of (1.) 
the fact, or series of facts, which constitutes the 
relation ; and (2.) the attitude or position in which 
the object denoted by the name stands to those 
facts. Two correlative names have the first part of 
the connotation in common ; they may, or may not, 
have the latter {i.e. they may or may not occupy 
an identical attitude with reference to the facts im- 
plied in the relation, but, in either case, they must 
both connote those facts). 

[Tims parent — offspring both connote the facts which connect 
them, but they each connote a different attitude or posi- 
tion in regard to those facts ; while consort implies another 
consort, both objects holding the same position to the facts 
which connect them.] 

12. Non-relative Names include, of course, all but 
relative names. 

13. Univocal, Equivocal, and Analogous Names. 

These are not different kinds of names, but merely dif- 
ferent modes of employing them. 

A name is used univocally when it is used in the 
same sense. 



THE CATEGORIES. 15 

A name is used equivocally when it is used in dis- 
tinct senses. 

A name is used analogically when it is used in a 
signification somewhat similar to its primary and 
proper meaning. 



CHAPTER III. 

The Categories. 

Mr Mill first examines and criticises the arrange- 
ment of the Categories given by Aristotle and the 
majority of Logicians; and afterwards proposes an 
arrangement of his own. 

The word " Thing " is used in its widest sense through- 
out this chapter. It is not limited to material or 
really existing objects, but includes every object of 
Sense or Imagination of which we can become con- 
scious, or to which a name can be given ; thus God, 
a spirit, a centaur, an attribute (as blueness, hard- 
ness), or a feeling (as oipain, anger), are all included 
amongst " Things." 

1. The Categories are an enumeration of the 
great classes under one or other of which every 
nameable thing may be included. 

[Or they are : — 

1. "The most extensive classes, or summa genera, 
into which all things can be distributed." 



16 THE CATEGORIES. 

Or— 

2. " So many highest predicates, one or other of 
which was supposed capable of being affirmed 
with truth of everything whatever."] 

The Aristotelic Categories are: — 



(1.) Substantia. 


(2.) Quantitas 


(3.) Qualitas. 


(4.) Relatio. 


(5.) Actio. 


(6.) Passio. 


(7.) Ubi. 


(8.) Quando. 


(9.) Situs, 


(10.) Habitus. 



(" Habitus" according to Sir "William Hamilton, expresses 
the relation of the container to the contained ; the 
idea being that of a man contained in his garments.) 

Mill's criticism of the Aristotelic Categories : — 

1. The list is unphilosophical and superficial; being a 

mere catalogue of the distinctions rudely marked 
out by the language of familiar life, without any 
attempt to penetrate to the rationale of even these 
common distinctions. 

2. It is Redundant; Actio, Passio, Ubi, Quando, Sikis, 

and Habitus are cases of Kelation ; Situs and Ubi 
are the same, viz., position in space. 

3. It is Defective; having no head, or summum genus, 

under which States of Consciousness can be classed. 

£It may be remarked, however, that good authorities maintain 
that Aristotle's classification was mainly grammatical, and 
was not intended by him as a list of the Categories in the 
usual Logical sense of the word. Thus the substantive is 
represented by "Substantia;" adjectives of quality by 
" QuaUta*;" of number by " QuantUas ;" of comparison by 
" Relatio;" adverbs of time and place by " Quando" and 



THE CATEGORIES. 17 

" Ubi;" and the different forms of the verb by "Situs," 
"Habitus," "Actio," " Passio." It is thus evident that 
distinctions in the grammar of words more or less underlie 
this arrangement ; and its appearance in a logical treatise 
may be accounted for by remembering that in Aristotle's 
day there was not that sharp line of demarcation between 
Logic, Metaphysics, and Grammar which exists now.] 

II. Kegarcling the Categories, then, as an 
enumeration of the classes of nameable things, 
Mr Mill gives first a Preliminary Enumeration 
of such classes ; and, afterwards, by re-arrang- 
ing and giving a more philosophical form to 
this list, he constructs his Final (or Proper) 
Enumeration of the Categories. 

1. Preliminary list of Classes of Things : — 

I. Feelings, or States of ^ ( c +• t? ■ *• 

_, J ' Sensations, Emotions, 

Consciousness or 



of Mind. 



II. Substances. 



Ideas, Volitions. 



I Occupying space ; the un- 

Bodies < known external Causes 

( of our Sensations. 

( The unknown precipient of 

j Minds <J feelings, or subject of 

so-called mental states. 



Quality. 
II. Attributes. <[ Quantity. 
Relation. 

IV. Certain Relations of\\ 
our Feelings. j j 



Coexistences, Sequences 
Similarities and Dis- 
similarities. 

B 



18 THE CATEGORIES. 

Respecting this enumeration it is only necessary to re- 
mark — (1.) That " Substance " here means anything 
which possesses attributes. Mind possesses attri- 
butes, properties, or qualities ; hence it is a " Sub- 
stance " in this sense. (2.) That Class IV., though 
merely including peculiar cases of Relation, is yet 
better placed apart, because the relations therein 
contained are peculiar, irresolvable and inexplicable. 
We know what we mean when we say one thing 
follows another, one thing coexists with another, or 
one thing is like another (we are speaking of ulti- 
mate resemblance), — we know what the state of 
mind is which accompanies the recognition of 
" Sequence," &c. ; but we cannot pretend to explain 
or analyse it. 

Mr Mill, however, goes on to show that Attributes are 
resolvable into States of Consciousness, and by in- 
cluding them in that Category, and treating " Bodies " 
and " Minds " as separate Categories, he arrives at 
the following proper and final 

2. Enumeration of the Categories : — 

I. Feelings or States of Consciousness (which includes 
Attributes). 

II. Minds — which experience those Feelings. 

III. Bodies — external objects which may excite certain 

of those Feelings. 

IV. Coexistences, Sequences, Similarities and Dissimil- 

arities of F< < Hugs. 

The principal point to be explained is, the resolving 
Attribute* into - of Consciousness or Feelings 

(a term used t<> designate any menial phenomena of 

which wc are conscious). How can an Attribute be 



I 



THE CATEGORIES. 19 

regarded as a Feeling ? To explain this we may say 
that evidently we know and can know nothing of 
Things, except through the Feelings they excite in us; 
the Attributes, or properties, of things are, in fact, 
only other names for the powers which those things 
possess of exciting certain feelings. To say that an 
object before me is "blue" is to say that a certain 
mental state is excited in me — a certain feeling — 
which I call sensation of blue (quality). To say that 
an object possesses Attribute " largeness" is to say 
that it excites a certain sort of feeling in me ; and 
so in every case of Attributes. That word properly 
means nothing more than the power of causing cer- 
tain sensations in our minds, — and these sensations 
are all that, at bottom, we can mean by Attribute. 
To say that I am conscious of the Attribute or 
quality " blueness " in an object before me, is 
nothing more than to say that I am conscious of a 
certain feeling or mental state, called a sensation of 
blue. A similar analysis applies to Attributes of 
quantity and relation as well as of quality. The 
distinctions, therefore, which we verbally make be- 
tween the properties of Things and the sensations we 
receive from them, must originate in the convenience 
of discourse, rather than in the real nature of what 
is denoted by those terms. 

[Doctrine of Berkely and the Idealist Meta- 
physicians — the non-existence of a material sub- 
stratum (" Noumenon ") in objects of Sense. 

Take any external object, as an orange : imagine 
its colour to be removed without being replaced by 
any other ; its odour, its taste, the sensations which 



20 THE CATEGORIES. 

it communicates by touch, and, in fact, the entire 
group of sensations which we can possibly have ex- 
cited in us by it, to be similarly dealt with — what 
would then remain ? Our ordinary conceptions would 
lead us to say the " thing itself/' the " nouruenon," 
the material substratum to which the properties or 
attributes belong. Berkely, on the other hand, would 
reply that nothing would be left, — that objects are 
nothing more than a bundle of sensations bound 
together by a fixed law ; and that a fixed law of 
connexion making the sensations occur together, 
does not necessarily involve a material substratum. 
Grant such a substratum to exist, and, in an object 
before us, to be instantaneously annihilated by 
Almighty fiat, the sensations being, however, pre- 
served unchanged, and how should we miss the sup- 
posed substratum ? Evidently we should know 
nothing of its absence, — the object would remain 
absolutely the same to us. 

Mill's remarks upon. — There is at least this much 
truth in this doctrine, that all we can know of 
objects is the sensations which they give us, and the 
order and connexion of those sensations. There is 
not the slightest reason for believing that the sen- 
sible qualities manifested by a thing are a type of 
anything inherent in the thing itself. An effect 
docs not of necessity resemble its cause; the sensa- 
tion of cold is not similar to anything in ice, nor of 
heat to anything in steam.] 



PE0P0SITI0NS. 21 



CHAPTEE IY. 

On Propositions. 

A Proposition is a sentence in which a pre- 
dicate is affirmed or denied of a subject. 

Nature and office of the Copula. — The Copula is 
the mere sign of predication — the sign of the con- 
nexion between Subject and Predicate. 

It does not indicate the actual existence of the subject ; 
the notion that it does so arises from the double use 
of the verb " to be " (viz., as a mere sign of asser- 
tion, and as equivalent to " to exist "). 

[Since the copula indicates the connexion between the two 
terms, whatever has to do with that connexion, rather than 
with either of the terms separately, may be regarded as 
belonging to the copula. See Negation and Modality.] 

Mill discusses the following distinctions in Pro- 
positions : — 

I. Affirmative and Negative. 

II. Simple and Complex (Conditionals, Disjunctives, 
&c.) 

III. Universal, Particular, and Singular. 



22 PROPOSITIONS. 

I. Affirmative Propositions are those in which 
the predicate is affirmed of the subject. 

Negative Propositions, those in which a predi- 
cate is denied of the subject. 

Hobbes Theory of Negation. 

Hobbes and some others state this distinction differently ; 
they recognise only the affirmative copula (is, are, 
&c), and attach the negative to the predicate (thus, 
man is not-mortal), and this with the idea of sim- 
plifying, by getting rid of, the distinction between 
affirming and denying, by treating every case of 
denying as the affirmation of a negative name. 

MiUs remarks on. — The distinction between affirming and 
denying is real, and is not to be got rid of by a 
verbal juggle. A negative name is merely one ex- 
pressive of the absence of an attribute, — so that when 
we affirm a negative name, we really affirm the ab- 
sence, not the presence, of anything; not that some- 
thing is, but that it is not. To put things together, 
and to put or keep them apart, will remain dis- 
tinct operations whatever tricks we may play with 
language. 

Modality, like Negation, belongs to the Copula ; 
it is not that we have indicated commonly so many 
different kinds of predicates, but so many different 
kinds of connexion between subject and predicate. 
For example — " The sun is rising," "The sun was 
rising/ 1 " The sun will be rising." Here we have not 
so many different kinds of " rising," but so many 



PROPOSITIONS. 23 

different ways in which "sun" and "rising" are 
connected. If a doubtful case should arise, as to 
whether a given part of a Proposition belongs to 
Copula or to Predicate, ask the question — Does it 
modify the meaning of predicate, or does it rather 
affect the mode or kind of connexion between the 
subject and predicate ? 

Propositions which merely assert a state of mind 
relative to a fact, and not anything directly in the 
fact itself, form a special class. They do not assert 
a connexion between two things, but what we think 
about that connexion. Thus, " Caesar maybe dead/' 
" C. is, perhaps, or probably dead," and such like, 
equal "lam not sure C. is alive," or " I am not sure 
that he is dead." 

II. Simple and Complex Propositions. 

A Simple Proposition is one in which one predi- 
cate is affirmed or denied of one subject. 

A Complex Proposition has a plurality of subjects, 
or of predicates, or of both. " Conditionals" and 
" Disjunctives " are the most important of this 
class. 

Conditional Propositions : — 

" If A is B, then C is D" is equal to this : — 
" The proposition C is D is a legitimate inference from the 
proposition A is BV 

The distinction, then, between a conditional and a cate- 



24 PROPOSITIONS. 

gorical proposition (as far as there is any distinction) 
is : — that a conditional proposition is a proposition 
concerning a proposition ; the subject of the asser- 
tion is itself an assertion. 

This property, even, is not peculiar to conditional pro- 
positions ; there are other sorts of assertions which 
may be made concerning a proposition, besides its 
inf erribility from something else. Like other things, 
a proposition has attributes which may be pre- 
dicated of it ; one of these attributes is its infer- 
ribility from another statement, as just said ; but 
many others frequently occur. Thus, " C is D is 
an axiom of mathematics ;" " C is D is a truth of 
Scripture, and a tenet of Protestants," &c, &c. 

The important position which conditional propositions hold 
in logical treatises is simply owing to this, — that 
what they predicate of a proposition, its being an 
inference from something else, is precisely that one 
of the attributes of the proposition with which Logic 
is most intimately concerned. 

[To prevent confusion, notice that Mill adopts "Whately's term 
"Hypothetical" as a genus including both " Conditionals 
proper," and "Disjunctives," — since Disjunctives are re- 
solvable into Conditionals.] 

Disjunctive Propositions : — 

Any one of these may be resolved into two or more con- 
ditionals, thus — 

" Either A is B, or C is D," is equal to 
«IfAfcnotB,CisD» j takent ether< 
" If C is not D, A is B," S 

A Categorical Proposition is one in which the 

assertion is direct, and not in any way conditional. 



PKOPOSITIONS. 25 

III. Universal Propositions, do. 

A general name is said to be distributed when it 
stands for each and every individual to which it can 
be applied as a name ; or, shortly, when it stands for 
the whole of its denotation. 

A Universal Proposition is one whose subject is 
distributed. 

A Particular Proposition is one whose subject is 
undistributed. 

A Singular Proposition has for its subject an 
individual name. 

(Not necessarily a Proper Name, but often one of those 
connotative or descriptive individual names which 
have been already mentioned. As, "The Founder 
of Christianity was crucified.") 

An Indefinite Proposition is one in which we know 
not whether it is Universal or Particular. They are 
only indefinite inform, the framer must know whether 
he meant it to be universal or not. 



CHAPTER V. 
The Import of Propositions. 

The main question discussed in this chapter is this, — What 
is the ultimate meaning of a Proposition 1 When we make 
an assertion, what is it of which we really speak, and what is 
asserted of those things of which we really speak 1 



26 PKOPOSITIONS. 

Since a Proposition consists of the Subject and Predicate, 
connected by the Copula, the main question naturally sub- 
divides into two (as above), which will be discussed under 
I. and II., while under III. some miscellaneous points will 
be examined. These two questions are : — 

I. Between what is a connexion asserted in a Proposi- 
tion ? in other words, What must we understand 
its terms to stand for or represent ? 

II. Having settled what the Subject and Predicate re- 
present, we next inquire, What kind of connexion 
between them may be asserted ? 

I. What is it of which we really speak in 
an assertion % What do the terms of a Pro- 
position stand for % 

In considering this question Mill first examines three 
views which have been advocated by different 
logicians ; and, after criticising these, he gives his 
own doctrine. 

\st Doctrine. — That a Proposition is the expres- 
sion of a connexion between two Ideas (i.e., that the 
terms must be understood to represent the correspond- 
ing ideas'). 

That is, — that a Proposition affirms or denies one idea 
of another; that "judging" is putting two ideas 
together, or bringing one idea under another, or 
ooxnparing two ideas, <>r perceiving the agreement 
or disagreement of two ideas. The whole theory of 
Propositions and of Reasoning being stated as if 
Ideas constituted essentially the subject-matter of 



PKOPOSITIONS. 27 

these operations. Thus, " Gold is yellow " — these 
logicians would say means that " My idea of gold 
includes or agrees with my idea of yellow." 

Mill's Criticism : — 

This doctrine is a serious mistake. It is true, as before 
stated, that an idea or notion of the things brought 
into connexion is a necessary preliminary to, or 
condition of, the intelligent assertion of a Proposi- 
tion ; but the assertion itself refers not to the ideas, 
but to the external facts. 

2d Doctrine. — That a Proposition expresses the 
relation between the application of two names. 

This was the doctrine of Hobbes : " In every Proposi- 
tion," says he, " the thing signified is the belief of 
the speaker that the predicate is a name of the same 
thing or things of which the subject is a name ; " 
in other words, a Proposition merely asserts that the 
two names which compose the terms have or have 
not been assigned to the same object or objects. 
Thus, " Man is a living creature," means that " liv- 
ing creature " is a name of everything of which 
" man ' ; is a name ; and if so, the Proposition is true ; 
if not, false. 

Mill's Criticism. — It may be remarked of this : — 

(1.) It is a property which all true Propositions possess ; 
it is, in fact, an assertion made in every Proposi- 
tion. 

(2.) It is the only analysis which is rigorously true of all 
Propositions without exception. 

(3.) But the only Propositions of which it is a sufficient 



28 PROPOSITIONS. 

account are that very unimportant class where 
both terms are proper names — as, " Tully is Cicero." 
(4.) Of all other Propositions it is a very imperfect 
analysis. Hobbes himself allows that general names 
are given to things, not accidentally, as it were, but 
because those things possess certain attributes, and 
it is strange that he did see that when we predicate 
of any subject a name given because of the posses- 
sion of certain attributes, that our object is not to 
affirm the name, but by means of the name to affirm 
the attributes. 

3d Doctrine. — That the terms represent classes. 

That is, that an assertion consists in including an indi- 
vidual or a class in another class, or excluding the 
one or the other from it. Thus, " Plato is a philo- 
sopher," means that Plato is included in the class 
" philosopher ; " " all men are mortal," that the 
class " men " is included in class " mortal beings." 
This theory is, in fact, framed as if nature had 
arranged all the objects of the universe into definite 
a priori classes ; things of the same kind being, as 
it were, done up together in parcels or bundles, 
called classes, so that it is sufficient for us to know 
in what parcel or class any given object is included, 
and we have then all the information as to its 
attributes which we can require. 

Mill's Criticism : — 

(1.) This doctrine is an example of va-rcpov nporcpou (last 
first) — or the logical error of explaining anything 

by something which presupposes it. The fact is, 
things are not found in nature marked out into 
defined and absolute classes. Classes are made by 



PROPOSITIONS. 29 

the human mind ; observing that certain objects 
have certain properties in common, we group them 
together into a class in virtue of these common at- 
tributes, and give them a name which connotes 
these attributes. The error in the theory, then, 
consists in this — objects are (in real fact) first put 
together into a class because they possess certain 
attributes, and then (according to this view) those 
attributes are inferred because the objects belong 
to the class. 

(2.) This theory is further essentially the same as that 
of Hobbes ; to refer anything to a class is the same 
as saying the class name is applicable to it; for 
both the name and the class are dependent upon the 
possession by the objects of certain attributes. 

(3.) This theory is the basis of the famous " Dictum de 
omni et nullo ;" the syllogism being resolved into an 
inference that whatever is true of a class is true of 
everything contained in that class. 

4th Doctrine. — Mill's view. — That a Proposition 
asserts the connexion between tvhat is connoted by 
the terms. 

That is, that every connotative term must be under- 
stood to stand for what it connotes ; the connoted 
attributes are the things between which in a Pro- 
position the connexion is asserted. A non-conno- 
tative term stands, of course, for its denotation, — 
i.e., the objects to which it is applicable. 

Thus, " Man is mortal," means " The attributes connoted 
by 'man' are always accompanied by attributes 
connoted by ' mortal.' " " Csesar is a man " equals 
the object "Caesar" possesses the attributes con- 
noted by name " man ; " and so in every other case. 



30 PROPOSITIONS. 

Instead of saying " attributes connoted by," we may sub- 
stitute "phenomenon" without altering the meaning. 

II. We now proceed to the second of the 
two questions, — as to what kinds of connexion 
may be asserted in Propositions. 

The matter of fact asserted in every (real) proposition is 
one or other of these : — 



Simple Existence. 

the fact of = Order in Time, 
the mode of = Order in Place. 



Coexistence 



Sequence. 
Causation. 
Eesemblance. 

~&\itfi?-st — Causation is only a case of Sequence, which is 
itself a case of Order in Time. Again, Coexistence 
is either Order in Time, that is Simultaneousness ; 
or Order in Place, i.e., arrangement, collocation ; 
as, " The boys stand in a straight row." We, there- 
fore, arrive at the following final arrangement : — 

1. Simple Existence. 

2. Order in Time. 

3. Order in Place. 

4. Eesemblance. 

1. Propositions asserting or denying the actual 

existence of something require no particular remark. 

They are such as — " There are such things as black 

swans," " White crows do not exist." 

This • 1 . i — . lb distinguished from the other three inasmuch as it 
docs Dot assert a connexion or relation of any kind between 
two things. 



PEOPOSITIONS. 31 

2 & 3. Propositions asserting the Conjunction (which 
includes Order in Time and in Place) in some way of 
two Phenomena, especially those which affirm or deny 
Coexistence or Sequence, are of great importance in 
Logic. 

Almost all Propositions expressive of physical facts, 
assert either the coexistence of two phenomena, or 
the sequence of one phenomenon after another. 
All Propositions asserting that one phenomenon is 
the cause of another, evidently assert sequence. 
The most important Propositions which assert 
Order in Place are mathematical loci ; that is, Pro- 
positions which affirm that a succession of points 
marked in a certain way will lie along a certain 
path. 

Propositions asserting Order in Time (Simultaneousness 
or Coexistence in Time, and Sequence) are so import- 
ant that, for the general purposes of Logic, they may 
be taken exclusively. Mill does this ; afterwards con- 
sidering separately those which assert Existence, and 
those which assert Resemblance, and Order in Place. 

It is evident that if one phenomenon is always conjoined 
with another, whether as simultaneous or as success- 
ive, that phenomenon becomes a mark of the other, 
i.e., whenever we meet with the one we may be 
sure of the presence of the other ; and every pro- 
position of conjunction comes within this formula — 
" One phenomenon is a mark of another phenomenon" 

4. Propositions asserting Resemblance (or Dis- 
similarity). 

There is a distinction in cases of Resemblance between 
phenomena which is important. We have — (1.) Ulti- 



32 PKOPOSITIONS. 

mate resemblance, where two things are like each 
other, but we cannot analyse that likeness, — it is 
simple, and incapable of being resolved into more 
elementary particulars ; and (2.) Resemblance in 
some assignable respects. Thus, if I say " This horse 
resembles other horses," I can enumerate the 
details of form, colour, &c, &c, in which the ani- 
mals agree ; but if I say this " Red is the same as 
that red," or " The hunger I feel to-day is like the 
hunger I felt yesterday," the resemblance cannot be 
thus analysed ; we can only say that there is a re- 
semblance, we cannot say in what it consists. It is 
evident, however, that any resemblance of the second 
kind between two things must consist of an aggre- 
gate of resemblances of the first kind. The ultimate 
resemblances are resemblances in our simple feelings 
(i.e., conscious mental states not compounded of more 
elementary states.) 
The most important Propositions asserting Eesemblance 
are those of Mathematics ; resemblance having then 
the form of equality or proportionality. 

III. Certain Miscellaneous Questions. 

1. Propositions having a general name as predicate 
do not properly assert resemblance. 

The contrary has been maintained by some ; they would 
say the assertion, "Gold is a metal," means " Gold 
resembles metals more than it does any other class 
of bodies." But a class is usually not founded upon 

a mere genera] unanalysable resemblance, but upon 
• mblance in certain assignable attributes com- 
mon to all its members. These attributes are con- 
noted by the class name, and it is the possession of 
these by the subject which we mean to assert, and 



I 



PROPOSITIONS. 33 

not mere general similarity. It might still be said, 
" Gold is a metal," i.e., " the attributes connoted by 
name gold are accompanied by attributes connoted 
by name metal," if no other metal existed ; just so 
we may say, "The Lord is God," though there is 
none other beside Him. 

There are, however, two exceptional cases, i.e., 
cases tuhere mere resemblance is predicated by a 
general name : — 

(a.) Where an individual is put into a class simply be- 
cause it resembles the members of that class more 
nearly than of any other, and not, as usual, because 
it possesses the distinctive attributes of that class. 
Thus we say, "Arsenic is a metal" though it differs in 
many respects from the other metals, because it re- 
sembles them more nearly than it does any other 
class of the elements. 

(6.) Where the class corresponding to predicate name 
consists of members which have only an ultimate 
resemblance, and not a resemblance in specific assign- 
able particulars. 

The classes in question are those into which our 
simple feelings may be divided — " white," " red," 
" bitter," " sweet," &c. If I say " This tastes bitter," 
at bottom I only mean that it resembles other tastes 
which I have previously known under that name, 
and it would not be really understood by any one 
who had never experienced a bitter taste. 

2. Propositions whose terms are abstract require 
no separate remark. They may always easily be 
changed into Propositions with corresponding con- 
crete terms. The abstract name denotes the attri- 

c 



34 VERBAL AND REAL PROPOSITIONS. 

butes which the concrete connotes; "humanity is a 
mark of mortality," = " man is a mortal being." 

3. Negative and Particular Propositions only re- 
quire a slight corresponding alteration in the expres- 
sion, thus : — 

u Jfo hwses are web-footed" = " the attributes connoted by 
horse are a mark of the absence of those connoted 
by web-footed." 

"Some birds are web-footed," = " attributes connoted by 
bird are sometimes accompanied by attributes con- 
noted by web-footed." 

It may be here remarked, once for all, that we consider 
the absence of an attribute as being itself an attribute. 
Thus, e.g., we call "not blue" an attribute. It is a 
convenience which avoids circumlocution. 



CHAPTER VI. 

Verbal and Real Propositions. 

Verbal Projiositions ( = Essential = " Ana- 
lytic " of Kant) are those in which the conno- 
tation of Predicate is part or whole of the 
connotation of the Subject. 

To these we may add those Propositions in which both 
predicate and subject are proper names. 

\ Verbal Proposition asserts, therefore, of a thing only 
what has already been implied when we uttered the 



VERBAL AND REAL PROPOSITIONS. 35 

name of the thing. To say " Man is an animal " 
conveys no information, " animal " being part of the 
very meaning of the word " man ; " for we should 
certainly not call anything not animal " man." 
Verbal Propositions, therefore, do not properly assert 
matters of fact, but only inform us as to the meaning 
of names. 

Peal Propositions ( = Accidental = " Syn- 
thetic " of Kant) are those in which connota- 
tion of Predicate forms no part of connotation 
of Subject. 

All Propositions, therefore, which have a proper name 
for the subject (and not for predicate also) — since 
such names connote nothing — come under this class. 
If the subject be a general name, it is, as already 
said, necessary that its connotation should not in- 
clude that of predicate. Thus — "Man is a being 
which cooks its food ; " " cooking his food " is not 
. included in the meaning of " man." 

II. Assumption in Propositions of the Peal 
Existence of the Subject. 

Take this Proposition "A is B " — under what 
circumstances are we to understand from the Pro- 
position itself that A actually exists ? 

1. Verbal Propositions do not, in strictness, imply that 
the subject really exists ; they are, as we shall see, 
really more or less perfect definitions unfolding the 
connotation or meaning of a name ; and for the 
ordinary copula "is" we may substitute "means, 



36 VEKBAL AND REAL PROPOSITIONS. 

without altering the assertion. This is clearly seen 
where the subject connotes a group of attributes 
brought together by the imagination only ; thus — 
"A centaur is a being half man half horse,"="A 
centaur means a being, &c." 

Nevertheless, as a matter of fact, most Verbal 
Propositions do involve a tacit assertion of the real 
existence of the subject. In such cases the appar- 
ently simple assertion really consists of a definition 
together with a postulate ; thus — " A triangle is a 
three-sided figure," = " Triangle means three-sided 
figure, "-f-" Such a figure exists ;" the last being the 
postulate or assertion of actual existence. The im- 
portance of this will be seen hereafter. 
2. Real Propositions do necessarily imply the real exist- 
ence of the subject, because if the subject be non- 
existent there is nothing for such a proposition to 
assert. 

Apparent exceptions may occur to this — thus we 
may say, " A ghost haunted his bedroom" " The gods 
dwell on Olympus ; but even here it is evident that 
the actual existence of "ghost" and "gods" is pro 
tempore assumed. 

III. Uses of Verbal Propositions. 

1. With the assumption or postulate of the actual exist- 

ence of the subject : — 
(a.) They may convey information of that existence as 

a fact. 
(b.) They may .serve as the basis of deductions, as do 

the axioms of mathematics. 

2. Without that assumption — that is, in strict accuracy 

— they only serve as definitions, — i.e., to unfold 
meaning or connotation of the subject. 



CLASSIFICATION AND PREDICABLES. 37 

With the exceptions here mentioned, Real Propositions 
are the only propositions which can convey any in- 
formation as to matters of fact, or from which any 
inferences as to matters of fact can be drawn. 

IV. Two modes of stating import of a 
{Real) Proposition. 

(Mill here, as usual, takes Propositions asserting conjunction 
as the Propositions of Logic. See p. 31.) 

(1.) Attributes connoted by subject are accompanied by 
attributes connoted by predicate ; or 

(2,) Attributes connoted by subject are marks of, &c. 

These are really equivalent, but (1.) points to the Pro- 
position regarded as a mere piece of knowledge; 
while (2.) points to its practical use. 



CHAPTEK VII 
I. On Classification ; and II. The Pbedicables. 
I. Classification. 

(Classification as a scientific process is discussed in Book IV. 
Here only a few elementary points are noticed.) 

1. Connexion of Naming and Classification. 

General nam.es have a meaning quite irrespective of 
classes (that is, when we assert a general name, as 
" man," of a subject, we mean to assert that that 



38 CLASSIFICATION AND PREDICABLES. 

subject possesses certain attributes, " animal y, 
" rationality," upright form," &c, &c, which has 
nothing to do with classes), but there is a twofold 
connexion between them : — 
(a.) Classes mostly owe their existence, as classes, to 
general names. 

It is perfectly evident that if we invent a name, 
connoting certain attributes, we thereby create 
a class, — consisting of everything which pos- 
sesses those attributes. Thus, suppose I take 
two attributes, "perfect molecular mobility" 
and " inelasticity," and devise a name " liquid," 
which shall connote or mean those properties, 
a class is ipso facto formed containing all ob- 
jects possessing those two properties. Theor- 
etically it matters not whether the objects are 
many, one, or none at all, — the class then being 
wholly imaginary. 
(b.) But sometimes, on the other hand, general names 
owe their existence to classes. 

A name is sometimes introduced because we have 
found it convenient to create a class. This is 
usually the case with the classes of Plants and 
Animals ; as when we divide Plants into " Phae- 
nogamous" and " Cryptogamous," &c. 

^ m 7 • 7 /• ™ f ft ea l kinds- 

2. Two kinds of Classes- j Not _ real ldnds ^ 

It is a fundamental principle in Logic that the power of 
framing classes is unlimited, as long as there is any, 
even the smallest, difference to found a distinction 
upon. Thus, out of the class " Man" we may cut a 
class " Christian," out of this a class " Protestant," 
again a class " .Bishop," out of " Bisbop" the class 
" black-haired," and so on ad infinitum. 



CLASSIFICATION AND PREDICABLES. 39 

But if we contemplate our classes when formed, we dis- 
cover that they constitute two very broadly distin- 
guished divisions,— Real hinds and Not-real kinds. 

Real hinds are classes the members of which are 
characterised by the possession of an inexhaustible 
number of common properties, not referrible to any 
common cause. 

Hence the differences between two individuals belonging to 
two distinct Real kinds are as innumerable as the points 
of agreement between two individuals belonging to the 
same Real kind. 

Not-real kinds are classes the members of which 
agree only in certain particulars which can be num- 
bered, — that is, which have only certain specific and 
determinate common properties. 

Thus, compare the class " Animals" with the class 
" White things; in the latter the members are not 
distinguished necessarily by any common properties 
except " whiteness" and any properties effects of 
"whiteness;" but a hundred generations have not 
exhausted the properties which are common to all 
" animals ;" and though physiologists are continually 
discovering new ones, yet there is no probability 
that they will ever be able to say that they know 
them all, — that they have arrived at a knowledge of 
every property which exists in "animals." More- 
over, these common properties are not referrible to 
any one cause. " Animals" therefore, is a Real 
kind ; " white things" is not. Similarly in chemistry, 
the name of any element represents a class of ob- 
jects. " Sulphur" for instance, is a class including 



40 CLASSIFICATION AND PREDICABLES. 

every separate piece of sulphur. Will the time ever 
come when chemists can say that they have ex- 
hausted the whole list of the properties which those 
different pieces of sulphur possess in common ? 
" Sulphur" like the other elements, therefore, is a 
Real kind. 
It is sometimes said that Real hinds are natural classes, 
or classes formed by nature ; we have already seen that 
classes are really framed by the human mind, but 
the expression is true thus far, — (1.) that Real kinds 
are classes for the recognition of which, as such, no 
elaborate process is generally required, because each 
of them is marked off from all others, not by some 
one or few properties which may be difficult to 
detect, but by its properties generally, by its tout 
ensemble ; and, further, (2.) the ends of classification 
would be subverted if we did not recognise Real 
kinds as classes. 

II. The Predicables. 

The Predicables are a fivefold division of general 
names, to express the different kinds of class rela- 
tion which may exist between the subject and pre- 
dicate of Propositions. 

Thus, take any Proposition, as " all A is B," — the sub- 
ject " A" represents a class or an individual, also 
predicate, being a general name, stands for a class 
" B," — the Predicables express the different kinds of 
relation which may exist between these two classes, 
or between an individual and a class. It is evident 
that there are no names which are exclusively 
genera, or exclusively species, &c, but that a name 
may be one or other, according to the subject of 
which it is predicated. 



CLASSIFICATION AND PREDICABLES. 41 

The five Predicables are : — 

CxGIlUS. 1 ( Distinctions founded upon the Nature of 

Species. f ( Things, Genus and Species being Real kinds. 

Differentia. 

Proprium. 

Accidens. 



Distinctions founded on Connotation 
of Names. 



A Genus is any Keal kind which contains the 
subject, but which, at the same time, is divisible into 
lower {i.e., less extensive) Keal kinds, in one of which 
the subject is also contained. 

Such a kind is a genus to all kinds below it ; a species to 
all kinds above it. 

" All kings are animals," — here " animals" is a genus to 
" kings," since being a Real kind it includes a lower 
Real kind " man," to which also " king " belongs. 

A Species {i.e., an infima species) is the proximate 
or lowest Eeal kind to which the subject can be 
referred. 

It is easily seen how these two distinctions are founded 
upon the nature of things, since whether a class is a 
Real kind or not, and whether it is the lowest Real 
kind in reference to a given subject, are clearly 
questions pertaining to the nature of things. Take 
classes "negro" and "white man," — now if these 
differ only in some few assignable peculiarities, they 
do not form distinct Real kinds ; but if they differ 
as a man and a horse differ, or as phosphorus differs 
from sulphur, in an inexhaustible number of particu- 



42 CLASSIFICATION AND PREDICABLES. 

lars, they are distinct kinds. Now, when we say the 
"Hottentots are negroes," whether "negro" is the 
species to Hottentot depends upon whether " negro" 
is a Real kind or not, — a question of the actual facts 
of nature. 

Differentia (in strictness) is the surplus of the 
connotation of the name of the species, over and 
above that of the name of the genus. 

A species (as man) may be referred to a different Genera 
{animal, vertebrate, mammal, &c), according to our 
purpose on the particular occasion. In any such 
case, it is evident that the name of the species will 
always connote all the attributes connoted by the 
name of the Genus, and also an excess of attributes 
peculiar to itself. Thus " man," besides connoting 
all that " mammal " connotes, connotes also a num- 
ber of other attributes peculiar to itself (erect form, 
rationality, &c, &c). It is this excess of connoted 
attributes which distinguishes " man " from all other 
species of Genus " mammal," — in other words, forms 
its Differentia in respect of that Genus. 
The Aristotelic Logicians, instead of taking the whole 
of such excess attributes, fixed upon that one of them 
which seemed to them the most important ; thus, 
in the case of " man " they picked out " rationality " 
as the Differentia. Since this usually fulfils the pur- 
pose for which a Differentia is framed, i.e., to dit- 
t&nguisih the species, we may give a 

Looser and more practical definition of Differentia, 
thus : The definition of a species is that part of the 
connotation of the specific name which distinguishes 



CLASSIFICATION AND PREDICABLES. 43 

the species in question from all other species of that 
particular genus to which we are referring it. 

("That part" being composed of one or more, or all, 
excess attributes.) 

[Special or Technical connotation. 

1. In names already connotative, we may, if our purpose 

require it, change the connotation, — that is, we may 
select from amongst the common properties of the 
objects denoted by the name, a set distinct from 
those previously assigned to the name. Thus 
"man" ordinarily connotes animality, a certain 
form, and rationality ; but in a natural history 
system it may be made to mean a certain arrange- 
ment of teeth, and the possession of two hands. 
This forms a special connotation of name " man." 

2. Even non-connotative names may have a connotation 

assigned them in this way ; thus, " whiteness " pro- 
perly connotes nothing, but it may be made to mean 
" a mixture of the different tints of the spectrum," 
and similarly in other cases.] 

A Proprium of a species is any attribute which, 
although not itself connoted by the specific name, 
yet follows from some attribute which that name 
does connote. 

There are tivo classes of Propria : — 

1. Those which follow by way of consequence, i.e., as a 
conclusion from premisses. 

Thus, the attributes connoted by name " paral- 
lelogram " are — "having four straight sides," and 
"having opposite sides parallel;" a proprium (of 



44 ON DEFINITION. 

species "parallelogram"), "having opposite sides 
equal," follows by way of co7isequence. 
2. Those which follow by way of effect, i.e., as an effect 
from a cause. 

Thus, " man " connotes, amongst other things, the 
attribute "rationality," from this a proprium of 
" man " " capacity of using language " follows as an 
effect. 

An Accidens is an attribute which, being neither 
connoted by the name of the species, nor following 
from any attribute so connoted, is yet found in the 
species. 

Accidents are of two kinds : — 

1. Inseparable — those which might, as far as we can see, 

be absent from the species without making it a dif- 
ferent one, and yet never are so absent. As, " black- 
ness " of species " crow." 

2. Separable — those which are sometimes present, some- 

times absent in individuals of the species. As, " red- 
haired " of species " man." 

A fortiori, these accidents are " separable" which 
are not even constant in the individual, as "being 
clothed," " being ill," &c. 



CHAPTER VIII. 

On Definition. 

A Definition is a Proposition declaratory of 
the meaning of a word ; or, more precisely, it 
is the statement in words of the constituent 



ON DEFINITION. 45 

parts of the facts or phenomena of which the 
meaning of every word is ultimately com- 
posed. 

Place in Logic and Use. — Definition is the Logical in- 
strument of the first division of the Science, that 
relating to Terms. It remedies their indistinctness, 
by giving a precise and fixed meaning to every name 
capable of having such a meaning assigned to it, so 
that we may know precisely what attributes it con- 
notes, and what objects it denotes. It is only in 
this way that our assertions can have a fixed and 
determinate import. 

Definitions are either : — 

I. Perfect 



II. Imperfect. < 



Incomplete Definitions. 
Accidental Definitions. 



Perfect (= Complete, Adequate, Scientific) Defini- 
tions are those which declare the whole of the facts 
which the name involves in its signification ; that is, 
in the case of connotative names those which unfold 
the whole connotation. 

Imperfect Definitions are those which do not do 
this. 

(These distinctions apply more or less to the definition 
of every name, but since the only non-connotative 
names which can be defined are the names of single 
attributes (see p. 12), which are defined by assign- 
ing the fundamentum of the attribute, this class 



46 



ON DEFINITION. 



may be at once dismissed. Of connotative names 
(see also p. 12), the class of primary importance 
here is that of concrete general names; and this 
chapter may be considered as dealing with these 
exclusively, unless otherwise notified. All that it is 
necessary to say concerning connotative abstract 
names (names of groups of attributes, or of attri- 
butes which have attributes) will be given separately 
under that head). 



I. A Perfect or Complete Definition is one 
which expresses the whole connotation of the 
Name. 



Form of a Definition. — The Definition must give 
the several attributes connoted by the name defined ; 
now these attributes may either be enumerated singly 
and seriatim, or several may be grouped together 
under one word. In either case, again, we may 
name the attributes either directly by their own 
proper abstract names, or indirectly by using words 
which connote them. Thus, 
ment : — 



we get this arrange- 



(1.) Attributes enume- 
rated singly. 

(2.) Attributes enume- 
rated in groups. 



in either 



'(a.) The attributes may be 
expressed directly by 
the abstract names 
which denote them ; or 

(b.) The attributes may be 
expressed indirectly, 
by names which con- 
note them. 



i 



ON DEFINITION. 47 

II. Imperfect Definitions are either : — 

1. Incomplete ) Name defined by part only of its con- 
Definitions. ) notation. 

2. Accidental 



Definitions 

or 
Descriptions. 



(Definition composed of some attribute 
which is no part of the connotation 
of the name defined. 



Imperfect Definitions are framed with reference to one 
practical use of Definition, — the discriminating the 
things denoted by the name from all other things, — 
rather than with regard to scientific accuracy. 

1. Incomplete Definitions serve the practical pur- 
pose of Definitions when it happens that all objects 
which possess the enumerated attributes possess 
those also which are omitted. 

Logical Rule that Definition should be " per 
genus et differ entiamy 

Incomplete Definitions seem to have been had in 
view by Logicians when they laid down this well- 
known rule. It may be observed of it : — 

1. That it would be better expressed " per genus et dif- 

ferentias" as it might then yield a complete Defini- 
tion. 

2. It is impossible thus to define all names capable of 

being defined, — summa genera, for example. 

3. The object aimed at by those who laid down this rule 

is unattainable ; they seemed to imagine that the 
function of Definition is to expound the division of 



48 ON DEFINITION. 

things into Keal kinds, and to show the position 
which each kind holds in reference to other Real 
kinds ; but this is impossible. 

2. Accidental Definitions or Descriptions. 

Are Definitions composed of any attribute or combina- 
tion of attributes which (though they are not con- 
noted by the name defined) happen to be common 
to the whole of the subject, and peculiar to it. 
Thus : — " Man is a self-clothing animal." 
"Man is a food-cooking mammal." 

It is only necessary to a Definition of this kind that it 
should be convertible with the name it professes to 
define ; that is, it should be predicable of everything 
of which the subject is predicable, and of nothing 



[Special or Technical Definitions. 

Such accidental definitions may be raised to the rank of 
incomplete or even of complete Definitions, by 
making the elements of the description part or all 
of the connotation of the name defined. 

Thus : — " Man is a two-handed mammal " is Cuvier's 
definition of "man," — what with him the name 
"man" actually connotes.] 

III. Definition of Abstract Names. 

1. Connotative Abstract names (viz., names of groups of 
attributes, or of attributes which have attributes) 
may be defined like concrete names, by enumerating 
the attributes which they connote ; the definition in 
fact being parallel to that of the corresponding con- 
crete terms. 



ON DEFINITION. 49 

Thus : — Humanity = Corporeity, Animality, rationality, 
erect form. 
A human being = A corporeal, animated, rational, 
erect being. 

2. Non-connotative abstract names {i.e., names of a single 
attribute) must be defined by analysing the funda- 
mentum of that attribute ; i.e., by enumerating the 
facts or phenomena which the attribute represents. 

Thus : — Eloquence =the faculty of influencing the affec- 
tions of men by means of language. 

IV. What Names can and cannot he de- 
fined. 

1. Every name whose meaning can be analysed can be de- 

fined, — whether concrete or abstract ; that is, every 
name in reference to which we can distinguish into 
parts, the attributes or set of attributes which form 
its signification. 

[Even when the fact or phenomenon is one of our 
simple feelings, and, therefore, incapable of analysis, 
the names both of the object which excites, and of 
its attribute or property of exciting the feeling, may 
be defined by saying that they do so, but the name 
of the feeling itself cannot be defined. Thus : — 

White thing = an object which excites the sensation of 

white. 
Whiteness = property of exciting the sensation of 

white.] 

2. The names which cannot be defined are : — 

1. Proper names — since they have no meaning. 

2. The names of our simple feelings, because their 
meaning cannot be analysed. 

D 



I 



50 ON DEFINITION. 

Thus the names " sensation of white," "sen- 
sation of pain," of "weariness," of "hunger," 
&c, &c, only mean similarity to sensations we 
have previously been accustomed to call by 
those names ; and if we wished to convey a 
notion of them to another we could only do so 
by calling up something similar in his own ex- 
perience. Language is not adequate to explain 
colour to a man born blind ; the similarity to 
previous feelings which the names of sensation 
of colour connote not being appreciable by him. 

V. Doctrine that Definitions are either of 
Names or of Things. 

Many Logicians have divided Definitions into two 
classes, thus: — 

1. Nominal Definition^ Definition of a name, i.e., explain- 

ing the meaning of a name. 

2. Real Definition = Definition of a thing, i.e., explain- 

ing the nature of a thing. 

En truth, all Definitions are definitions of names, and of 
names only, but what they, confusedly perceived, 
and therefore vaguely indicated, is really the dis- 
tinction, not between definitions as such, but be- 
tween definitions without, and definitions with, a 
postulate or assumption of the real existence of things 
corresponding to the name defined. (See p. 30). 

1. Definitions which merely declare the meaning of a 
name, without any assumption as to real existence 
of the subject. This they called " Nominal Defini- 
tion^ or definition of a name. Such declare nothing 
as to matters of fact properly so called, but only the j 



ON DEFINITION. 51 

meaning which custom has assigned to a name, and 
therefore no conclusion as to matters of fact can be 
drawn from them. 

2. Definitions which explain the meaning of a name, but 
at the same time assume the real existence of the 
subject. This they termed "Real Definition" or 
definition of a thing. 

If in the former case we put " means " for " is," no change 
is made in the meaning of the proposition, nor is 
any inference which we may draw from it (being 
only inferences as to meaning of names) affected. 

In the latter, the change leaves the true definition, but 
withdraws the implied assertion or postulate of real 
existence ; and it is self-evident that no matter of 
fact could be inferred from a proposition which 
merely declares the sense in which we employ a word. 

As further arguments in support of the view that 
an apparent inference from a Definition is really an 
inference from the contained postulate, Mr Mill 

1. Examines Euclid I. 1, and shows that the very first step, — 

"about centre A with radius A B, describe a circle," 
depends upon the possibility of the actual existence of a 
circle ; 

2. And further shows that if the inference is really from the 

Definition, we may draw a false conclusion from true pre- 
misses — A dragon is a thing breathing flame, a dragon is a 
serpent : therefore, some serpent breathes flame. Here, 
in the major premiss, the definition is true, but the pos- 
tulate is false, and the conclusion being false shows that 
our inference is really from tbe false postulate, not from 
the true definition. 

[That definitions are really the premisses of scientific in- 
ference (as in Euclid) is sometimes defended by say- 
ing that they are so, provided that they are framed 



52 ON DEFINITION. 

conformably to the order of nature — that they assign 
such meanings to terms as shall suit actually exist- 
ing objects. From the meaning of a name, we are 
told, it is possible to infer physical facts, provided 
the name has corresponding to it an existing thing. 
But from which, then, is the inference really drawn 
— from the existence of a thing having the pro- 
perties, or from the existence of a name meaning 
them ?] 

These postulates are not always exactly true, — 
the things whose existence they tacitly assert do not 
always exist exactly as defined. 

Thus, probably no circle ever existed whose radii were 
precisely equal — no point without magnitude, no line 
without breadth, &c. 

Hence these postulates are really suppositions or hypotheses; 
that is to say, we suppose such points, lines, circles, 
&c, to actually exist as described in the definitions, 
though they do not in fact. 

This explains what Mill means by saying that "geometry 
is based upon hypotheses or suppositions ; " which is 
evidently true, because we have already seen that 
its inferences (putting the axioms aside for the pre- 
sent) are really from the postulates in the defini- 
tions, and these postulates or tacit assertions are, in 
fact, suppositions. 

Since mathematical truth (at least in part) rests ulti- 
mately upon such suppositions, a difficulty arose in 
understanding how the most certain of all truths 
could rest upon premisses which, so far from being 
exactly true, wen: certainly not true to the whole 
extent assumed. The real answer to this difficulty 
is, that as much of the premisses is true as is required 
to support as much as is true of the conclusion. 



ON DEFINITION. 53 

Some get over this apparent difficulty by saying 
that the Definitions of Mathematics are really De- 
finitions not of the things, hut of our ideas of the 
things. They say that what we argue about are our 
mental pictures or ideas of points, circles, lines, &c, 
which do exactly correspond to the Definitions. 

On this Mill remarks : — 

1. Even if we can form a mental picture of a mathemati- 

cal point, line, &c, the definition postulates the real 
existence of such an ideal picture, and from those 
postulates all inferences must really be made. The 
case is in fact exactly the same as before, only we 
have a mental picture instead of one drawn on an 
external surface. 

2. As a matter of fact, however, the mind can form no 

such pictures or notions as the hypothesis implies. 
We cannot conceive a line without breadth, &c. 
All that we can do is to attend to its length exclu- 
sively, neglecting the breadth for the time ; and so 
in every case we can attend to and deal with a single 
element, as existing alone, but we cannot picture it 
to our minds as actually so existing. 

Proper meaning of " Definition of a Thing." 

In Book IV. Mill remarks that if we retain this expres- 
sion in Logic, we must give it this meaning : — 

The Definition of a thing, or rather of a class of things 
(for we cannot define an individual), is defining the 
name in such a manner that it shall still continue to 
denote those things. 

VI. An inquiry into the Definition of a 
Name (i.e., as to what the Name should 
mean) — 



54 ON DEFINITION. 

Is an inquiry into the specific attributes in which the 
objects which the name usually denotes, agree or 
differ. A definition is therefore not arbitrary, and 
often involves a long examination into the proper- 
ties of things. 

Thus, suppose we wish to define "just actions" it would 
be necessary to collect instances of actions to which 
that designation is applied, and to examine them 
carefully to determine in what they agree. Having 
settled the properties, qualities, or attributes com- 
mon to all just actions, we have next to decide 
which of these are best adapted to enter into our 
Definition. 

This inquiry is, however, rendered difficult by : — 

1. The fact that words (particularly some abstract 

names) are constantly used without any definite 
connotation, except that of vague resemblance to 
other things called by the same name. 
Thus, when ordinary people speak of an act as "just" 
or "noble," they often really mean nothing more 
definite than a vague feeling that the act in question 
resembles others which they have heard so called. 

2. The transitive application of words. There is a con- 

stant tendency in men, when they meet with a new 
object, not to invent a new name for it, but to give 
it the name of some known object which seems to 
resemble it most. In this way a name may pass 
from object A to object B, from this to C, again to 
D, and so on till every vestige of definite significa- 
tion is lost, and the various objects denoted by the 
name come to have really nothing in common. The 
name M beautiful" is perhaps an example of this 
process. 



BOOK II. 
INFERENCE OR REASONING IN GENERAL. 



CHAPTER I. 

Inference in General. 

A Proposition is said to be proved when we believe it to be 
true by reason of some other fact or statement from which 
it is said to follow. 

I. Inferences improperly so called : — 

That is, the cases in which a Proposition, osten- 
sibly inferred from another, appears on analysis to 
assert merely the very same fact, or part of the same 
fact asserted in the first. 

1. (Equipollency or equivalence of propositions. Thus, 

"all men are mortal," for " no man is exempt from 
death." 

2. Sub-alternation — All A is B .\ Some A is B. 

3. Conversion — As some sovereigns are tyrants .% some 

tyrants are sovereigns. Both propositions assert 



56 INFERENCE IN GENERAL. 

the same fact — that the attributes connoted by 
"sovereign" are sometimes conjoined with those 
connoted by " tyrant." 
4. Repetition of connotation — i. e., when predicate of con- 
sequent is part of connotation of predicate of ante- 
cedent. 

As " Socrates is a man," «■■ " Socrates is an ani- 
mal." " Animal" being part of connotation or 
meaning of " man." 
£5. All other cases of so-called Immediate Inferences may 
be added to Mill's list. Such as (1.) Of Opposition 
— A is B .■. A is not non-B ; (2.) Of Relation — A is 
the son of B .•. B is father of A ; and such like.^ 

II. Logical forms of Inference. 

1. Induction is reasoning from particulars to 
generals ; or, more correctly, inferring a Proposition 
from Propositions less general than itself, — the con- 
clusion being more general than the largest of the 
premisses. 

2. Ratiocination (= Syllogism) is reasoning from 
generals to particulars ; or, more correctly, inferring 
a Proposition from Propositions equally or more 
general than itself, — the conclusion being less or 
only equally general with the largest of the pre- 
misses. 

3. Inference from particulars to particulars — that 
is. from individual cases to another individual case. 

If from having seen A, B, (J, and some other persons die 
(supposing, of course, that I know nothing except 
what I observe myself), I infer E also is mortal, I 
evidently reason from the former particular cases to 



INFERENCE IN GENERAL. 57 

the yet untried case. A little consideration will 
show us that a large number of our every-day infer- 
ences are of this sort. 

There is, however, no real difference between this 
mode of reasoning and Induction. For it is clear 
that, in order to be logically warranted in my con- 
clusion that E is mortal, I must have evidence 
enough from my antecedent particular cases to sup- 
port the inference " Any man is mortal," — for if the 
evidence did not amount to proving this, how could 
I be sure that E might not be amongst the excep- 
tions? But if it prove "any man is mortal," it of 
course proves " all men are mortal," — i.e., it is an 
Induction, or the inference of a general proposition 
from particular cases. As Mill says — "Whenever 
we are logically warranted in arguing from a set of 
particular cases to some new case, we are also war- 
ranted in inferring the general proposition, which 
includes that new case," and this being so, we may 
consider this mode of reasoning as identical with 
Induction. 
Further, then, since (as will be shown hereafter) Eatio- 
cination is only the interpretation and application of 
Inductions, all Logical Inference consists in Induc- 
tion, or in the interpretation and application of 
Propositions arrived at by Induction. 

Induction is without doubt a true process of infer- 
ence — the facts stated in conclusion are bond fide 
different from the facts given in the premisses. If, 
from examining four cases of mortality in man, I 
infer E also is mortal, — this last fact is clearly dis- 
tinct from the others ; still more if I lay it down 
that " all men are mortal," I assert innumerable 
separate facts, — X is mortal, Y is mortal, &c, &c. 



58 THE SYLLOGISM. 

CHAPTER II. 

The Syllogism. 

In the Analysis of the Syllogism Mill takes the two elemen- 
tary forms of the first figure, Barbara, Celarent, as the uni- 
versal types of all correct Ratiocination, — the first for affir- 
mative, the second for negative conclusions. An argument 
in any other figure may be reduced to one of these. 

I. Initial Analysis of the Syllogism. 

In both these general types the major premiss is uni- 
versal ; all Ratiocination, therefore, starts from a 
general Proposition. 

The minor premiss is affirmative, asserting that some- 
thing is contained in the class of which something 
has been affirmed or denied in the major premiss. 

The conclusion, then, infers that what is affirmed or 
denied of the entire class may be affirmed or denied 
of the objects asserted to be in that class. 

It was by regarding this analysis as sufficient and ex- 
haustive, that the Dictum de omni et nullo came to 
be accepted as the Axiom of Ratiocination. 

II. This dictum is a mere identical Pro- 
position, and not the fundamental Axiom of 
Syllogism. 

A (whole) class is, in fact, the same thing as (all) the 
individuals included in it. Therefore if we say — 






THE SYLLOGISM. 59 

" Whatever is true of a whole class is true of every 
individual in that class," it is the same as saying — 
" Whatever is true of all the individuals of a class 
is true of every individual in it," — an obviously 
identical Proposition. 
To give, therefore, any meaning at all to the Dictum, we 
must regard it, not as an axiom, but as a round- 
about definition of a class (i.e., an assemblage of in- 
dividuals of which the same Proposition is true). 

III. Fundamental Axiom of Ratiocination. 

" Whatever (A) is a mark of any mark (B) is a 
mark of that (C) which this last (B) is a mark of." 

That is — if A is a mark of B, and B a mark of C, then 
A is a mark of C. 

Another form of the Axiom of Syllogism is — 

" Whatever possesses any mark, possesses that 
which it is a mark of." 

That is — A possesses B, which is a mark of C, therefore 
A possesses C. This is really identical with the 
first, but a little varied in the expression. Either 
may be taken. 



60 FUNCTIONS AND VALUE OF SYLLOGISM. 



CHAPTEE III 
Functions and Value of Syllogism. 

I. Does the Syllogism really involve a Petitio 
Principii % 

(This question must be understood to mean — Does the con- 
clusion of a Syllogism assert any fact or facts bond fide 
new, and distinct from what has been asserted in major 
premiss ?) 

" It must be granted," says Mill, " that in every 
Syllogism, considered as an argument to prove the 
conclusion, there is such a " Petitio Principii" i.e., 
no really new fact is asserted in conclusion. This 
will be clearly shown by the analysis of the Syllogism 
itself, but the following is a summary of the argu- 
ments which bear on the question: — 

1. That the Syllogism is conclusive from its mere form, 

i.e., by a comparison of the language. Clearly this 
could not be the case if any new fact were asserted 
in conclusion. 

2. The accounts given by its defenders : — 

(a.) That it is vicious if the conclusion assert any- 
thing more than is asserted in the premisses. 
(b.) That its function is merely to prevent inconsist- 
ency in our opinions. 
:>,. An examination of the Syllogism itself shows that 



FUNCTIONS AND VALUE OF SYLLOGISM. 61 

both major premiss and conclusion are ultimately 
inferences from the same set of particulars. 
4. That the Syllogism is a process of interpretation is 
shown also by an examination of those cases where 
the major premiss is not derived from experience, 
as in Law and Theology. 

II. What, then, is the true nature of the 
Syllogistic process ? 

The following familiar explanation will, perhaps, 
give a clear view of this important point : — 

Suppose we have observed A, B, C, and D to be mortal ; 
let us assume that these particular cases constitute 
evidence sufficient to justify us in concluding that 
any new case, X, is mortal ; this is so far a case of 
inference from particulars to particulars. But it 
has been already explained, that in order to be war- 
ranted in inferring that any indifferent individual, 
X, is mortal, we must be warranted in inferring 
" All men are mortal " (for if not, our X might be 
amongst the exceptions) — i.e., any evidence which 
proves the particular Proposition must also prove 
the general. Thus : — 

_, _ _ „ , _ ( observed facts — the 

(1.) A, B, C, D, &c, are mortal j premisses or CTidence . 

(2.) All men are mortal (3.) X is mortal. 
(= attributes of man 
are a mark of attri- 
bute mortality). 

(2.) and (3.) Being two conclusions, either of which may 
be drawn separately without thinking of the other ; 



62 FUNCTIONS AND VALUE OF SYLLOGISM. 

but if the evidence prove one, it must also prove the 
other. 
But, further, instead of remembering the details of our 
evidence — the special particular cases from which 
our conclusion was drawn — we may, once for all, 
retain a record or memorandum of all that it will 
prove by simply bearing in mind the general Pro- 
position. Having once satisfactorily shown that 
" the attributes of man are a mark of mortality," we 
may dismiss from our minds the antecedent parti- 
cular cases which proved it, and remembering only 
the generalisation itself, may apply it to new in- 
stances as they arise from time to time, — the appli- 
cation consisting in the ascertaining that a new 
instance possesses the mark therein laid down (in 
the example — that the new object possesses the 
attributes of man, which are a mark of mortality). 
This process of applying a general proposition (the 
major premiss) constitutes the Syllogism ; which is, 
in fact, a kind of reasoning in which, in place of the 
evidence itself, we substitute a record or memo- 
randum of all that that evidence will prove, and 
then proceed to interpret and apply that record. 

From this it is evident: — 

1. That the conclusion is not really drawn from the 

major premiss, but according to it. The record or 
summary of the kind of conclusions we may draw 
from given evidence, must not be confounded with 
that evidence itself. 

2. That the Syllogism is not the mode in which we must 

reason, but only a mode in which we may reason. We 
may, and, indeed, constantly do, infer from observed 
individual cases to a new case, without ever think- 






FUNCTIONS AND VALUE OF SYLLOGISM. 63 

ing of a general proposition — as from (1) to (3). If, 
however, we do pass through the generalisation, the 
whole process will stand thus : — 

(1.) A, B, C, &c, are mortal ) An Induction leading to 
.v (2.) All men are mortal J a generalisation. 

(2.) All men are mortal \ Interpretation and applica- 
X is a man > tion of the generalisation = 

.". (3.) X is mortal J Syllogism or Deduction. 

All inference, therefore, is fundamentally from particu- 
lars to particulars, with the option of passing 
through a general proposition : and such a general 
proposition is a register or record of an inference 
already made, and a short formula for making 



III. Advantages of throwing the result of an 
Induction into the form of a general Pro- 
position : — 

1. The Induction may be made once for all. It is evident 

that having once proved the general Proposition, we 
need no longer trouble to remember the original 
particulars, but have only to apply the generalisa- 
tion to new particular cases as they arise. 

2. A general Proposition brings before the mind all that 

our evidence will prove, if it prove anything. 

3. It presents a larger object to the imagination — i.e., it 

is naturally felt to be of greater importance than 
any particular assertion, and hence — 

4. It tends to prevent us from being influenced by Bias or 

Negligence in our Inference. 

5. And, finally, it brings before us all possible parallel 



64 FUNCTIONS AND VALUE OF SYLLOGISM. 

IV. The use of the Syllogistic form, then, i.e., 
of passing through the general Proposition in 
our Keasoning, — is that it is a most important 
collateral security for the correctness of the 
Inductive process which gives us that gener- 
alisation. (See III. 2-5.) 

V. The use of the Syllogistic rules is to 
secure accuracy in the interpretation of the 
general Proposition. 

VI. Syllogism the test of Reasoning. 

The Syllogistic form is a test of the accuracy of the 
generalisation ; the Syllogistic rules are a test of 
the accuracy of the interpretation of the generalisa- 
tion. 

(If we substitute " security for" instead of " test o/," perhaps 
the meaning will be clearer.) 

VII. Syllogism not the universal type of 
Reasoning. 

Because, as already shown, we may reason without pass- 
ing through any general Proposition, — we may infer 
directly from observed partieulars to a new particu- 
lar case ; and in simple and obvious cases we habitu- 
ally do so. It is a matter of choice and convenience, 
therefore, whether we reason after the type of the 
Syllogism or not. 



FUNCTIONS AND VALUE OF SYLLOGISM. 65 

VIII. What is the universal type of Reason- 
ing f 

1. Certain individuals (having a certain attribute, A) 

have (also) a given attribute, B ; 

2. An individual or individuals resemble the former in 

possessing attribute A ; 

■•■ 3. They resemble them also in possessing given at- 
tribute B. 

This type does not claim to be conclusive from the mere 
farm; its validity in any particular instance must 
be determined by the canons of Induction. 

IX. Functions of Major and Minor Pre- 
misses in a Syllogism. 






1. The major premiss asserts something, — which has 

been found true of certain known cases, — to be true 
of all other cases resembling the former in certain 
given particulars. Or, 
That one phenomenon is a mark of another phenomenon. 

2. The minor premiss asserts that some new case re- 

sembles the former in the given particulars. Or, 
That some new cases possess the mark asserted in the 
major. 



[X. Dr Thomas Brown's Theory of Syl- 
logism. 

Dr B. dispensed with the major premiss, asserting- 
that the premisses in a Syllogism consisted of the 
minor alone, thus : — 

Socrates is a man ; 
■\ Socrates is mortal ; 

E 



66 FUNCTIONS AND VALUE OF SYLLOGISM. 

but allowing at the same time the necessity of per- 
ceiving the connexion between man and mortality, 
which is only another way of saying that we must 
have either actually or implied a major premiss, 
which asserts that connexion.] 

XI. Relation between Induction and De- 
duction. 

M Although, therefore, all processes of Inference in which 
the ultimate premisses are particular cases, whether 
we conclude from these particulars directly to a new 
case, or to a general proposition, according to which 
we afterwards conclude concerning new particulars, 
— are equally Induction, yet we shall consider 

Induction as belonging more peculiarly to the first step, 
the establishing the general proposition — that one 
phenomenon is a mark of another phenomenon, — 
while 

Deduction is the remaining operation, which is that of 
interpreting and applying that general proposition. 
And we shall further consider every process by 
which anything is inferred concerning a new case, 
as consisting of an Induction followed by a Deduc- 
tion ; because, though we need not carry on the 
process in this form, yet it may always be thrown 
into that form, and ought to be so when accuracy is 
essential." 

XII. Objections to Mill's Theory of Syl- 
logism. 

1. Whatelys:— 
It cannot be correct to say that a Syllogism is only a 



FUNCTIONS AND VALUE OF SYLLOGISM. 67 

special mode of dealing with the conclusion of an 
Induction, for the argument in every Induction is 
itself a Syllogism. Thus, I observe that " A, B, C, 
&c, are mortal," and from this I infer "all men are 
mortal ; " but in this Induction we tacitly assume 
that " what is true of A, B, C, &c, is true of all 
men," and our Inductive argument is in fact a 
Syllogism, of which this assumption is the major 
premiss. 

Mill replies ; — 

There is no such major premiss in the Inductive argu- 
ment as is here assumed ; in fact, the proposition 
which Whately gives as such is more properly part 
of the conclusion of that argument. Whenever we 
draw a conclusion from evidence, we, of course, 
tacitly assert the sufficiency of that evidence to 
support the conclusion, and therefore an implied 
assertion of the sufficiency of the evidence may be 
said to be part of the conclusion. It is just so here, 
— the supposed major is really an assertion of the 
sufficiency of the evidence, and is therefore neces- 
sarily involved in the conclusion. The premisses of 
the Induction are simply and only — " A, B, C, &c, 
are mortal ;" having observed these particular cases, 
we are led by an instinctive mental tendency to ex- 
pect, at least, that the same will be true of the next 
case, and, finally, after what we think a sufficient 
number of observed cases, that it will be true of all 
men. 

2. Others object that Mill's theory does away with 
the minor premiss. 

" If, say they, the major includes the conclusion, then we 



6$ TRAINS OF REASONING. 

could affirm the conclusion without the minor, 
which we evidently cannot do." 

MiU replies : — 

When we say that the major premiss includes the con- 
clusion (that is, that the conclusion asserts the same 
facts, or part thereof, as the major premiss), we do 
not mean that it individually specifies all it in- 
cludes. It only includes by giving marks ; " all red- 
haired men are choleric," does not name the indivi- 
duals, but it lays down a mark (" red-haired") by 
which we may know them, and we require a minor 
premiss to show us that a new case is included in 
the major by possessing the mark. 



CHAPTER IV. 
On Trains of Reasoning and Deductive Sciences. 

I. A Train of Reasoning ( = the "Sorites" 
of other Logicians) is a series of Inductive 
Inferences, from particulars to particulars, 
through marks of marks. 

[The formula of the ordinary Syllogism we have 
seen to be this : — 

1. (Attribute) A is a mark of (attribute) B (1st Induction). 
X possesses (attribute) A. 
.•. X possesses (attribute) B. 



TKAINS OF REASONING. 69 

Here we are supposed to know our minor by direct 
observation — by seeing or feeling that X possesses 
A ; and if so, to get the conclusion, we require but 
a single Induction, that which proves " A is a mark 
ofB." 

But it may happen that the fact that X possesses A 
is not obvious to the senses ; direct observation, 
perhaps, only shows us that X possesses C, and we 
require a second Induction to prove that " C is a 
mark of A," when we get the required minor of first 
Syllogism by a second Syllogism. Thus : — 

2. C is a mark of A (2d Induction) . 
X possesses C. 

.-. X possesses A (minor of 1st). 

But again it may happen that our senses do not inform 
us even that X possesses C ; by direct observation 
we may only know that X possesses D ; again we 
must go through a similar process ; that " D is a 
mark of C " must be proved by a third Induction, 
and we must then frame a third Syllogism before 
the minor of our second can be proved, and, there- 
fore, before our first conclusion can be logically 
established. Thus : — 

3. D is a mark of C (3d Induction). 
X possesses D. 

.\ X possesses C (minor of 2d). 

It is evident that this process may go on through any 
number of Inductions and Syllogisms ; and if we 
arrange the Syllogisms in an order exactly the re- 
verse of that here set down, they will be seen to 
form a " Train of Reasoning." The reason why it 
is said to be " through marks of marks " is obvious — 
X possesses D (directly observed), D is a mark of C, 



70 TRAINS OF REASONING. 

C is a mark of A, A is a mark of B, .\ X possesses 
B. If these Propositions be written down, each 
under the preceding, they will be seen to form the 
Sorites of ordinary Logicians.] 

A more complex form of Trains of Reason- 
ing is this type : — 

E is a mark of D. 
F „ of C. 
G „ of B. 

Also D C B is a mark of N. 
.-. E F G is a mark of N. 

II. Why do Deductive Sciences exist? or 
to phrase it more accurately — Why are there 
any difficulties in Deductive Sciences ? 

Sciences which employ Trains of Reasoning are Deduc- 
tive Sciences ; now each fresh step or Syllogism 
requires a separate Induction, and if these Induc- 
tions have been obtained, it would seem that very 
little difficulty would remain in merely framing the 
chain of Deduction. Yet we find very difficult 
Deductive Science may exist (as Mathematics) 
where the primary Inductions are of the simplest 
character. 

The explanation of this is, that (1.) There may be much 
difficulty in finding an Induction which compre- 
hends any given case ; and (2.) It often requires the 
highest scientific ingenuity so to combine Inductions 
as at length to reach one which will directly include 
our case. 

Mill illustrates this by proving Euclid I. 5, direct from 



TRAINS OF REASONING. 71 

the six primary Inductions which are necessary to 
that proof. 

III. Differences between Deductive and 
Experimental Sciences. 

The opposition is not between Deductive and Inductive, 
but between Deductive and Experimental. 

A Science is Experimental in proportion as every new 
case requires a new Induction, — i.e., a new set of 
observations and experiments for its elucidation. 

A Science is Deductive in proportion as every new case 
can be brought under an old Induction, i.e., in pro- 
portion as every new case may be elucidated with- 
out a new set of observations or experiments. 

The generic diference, then, between Deductive and Ex- 
perimental Sciences is this : That in the former we 
have, in the latter we have not, been able to dis- 
cover marks of marks. 

IV. How an Experimental tends to become 
a Deductive Science. 

Thus, in an Experimental Science, the Inductions lie 
detached — A is a mark of B ; C is a mark of D ; E 
of F, and so on. Now (1.) A new Induction may, at 
any time, connect two or more of these pairs by 
showing, for instance, that B is a mark of C, when 
we could infer without experiment that A is a mark 
of C, and even of D ; and (2.) Some new Induction 
may connect hosts of these isolated pairs at once, 
thus making the Science largely Deductive at a single 
stroke. It is, therefore, this class of discoveries which 
are most potent in changing a Science in this 



72 DEMONSTRATION. 

V. The grand agent, however, for trans- 
forming Experimental into Deductive Sciences 
is the Science of Number. 

The properties of number are alone, in the most rigorous 
sense, properties of everything whatever. But these 
truths can only be affirmed of things in respect of 
their quantity. Now, if it come to be discovered 
that variations of quantity, in any class of pheno- 
mena, correspond regularly to variations of quality, 
every formula of Mathematics which relates to 
quantity becomes so far a mark of qualities in those 
phenomena, and thus so far renders the Science 
Deductive. 

Thus, when in Geometry, it was shown that every 
variety in position of points, direction of lines, forms 
of curves, or of surfaces (all qualities), had a corres- 
ponding relation of size (quantity) between two or 
three co-ordinates, an unparalleled Deductive ex- 
tension was thereby given to that Science. 



CHAPTER V. 

On Demonstration and Necessary Truth. 

Necessary Truth, according to the common 
definition, is such as is supposed to be inde- 
pendent of the evidence of experience. 

Unless otherwise specified, we shall here always under- 






DEMONSTRATION. 73 

stand the expression in this sense, but Mill would 
define it thus : — 

Necessary Truth is such as necessarily fol- 
lows from assumptions which, by the condi- 
tions of the inquiry, are not to be questioned. 

What is really meant by necessity here, 
therefore, is certainty of Inference. 

I. The character of Necessity (and even, 
with some reservations, the peculiar certainty) 
attributed to Mathematical Truth is an illu- 
sion. 

As to their peculiar certainty, it only consists in this — 
that mathematical conclusions are not liable to be 
interfered with by counteracting causes. In them- 
selves they are not more certain or exact than of 
any other science ; for Induction being at the root 
of all science, the first principles of each, if validly 
inferred, must be equally exact ; and if the De- 
ductions are correct, the results must be equally 
certain. 

The necessity, too, of geometrical truths consists only in 
this, — that they necessarily follow from the granting 
of the primary suppositions (or hypotheses) from 
which they are deduced. Their necessity, in fact, 
is certainly of inference. 

These primary suppositions are : — 

(1.) Definitions with their contained postulates. 
(2.) Axioms. 



74 DEMONSTRATION. 

Some of these suppositions (viz., the hypothetical postu- 
late in the definitions — which assume the actual 
existence of things corresponding to those defini- 
tions) are not only not necessary, they are not even 
strictly true. (See p. 52.) 

Some others of the first principles of Geometry are 
axioms, which are absolutely true without any 
mixture of hypothesis or assumption. That " the 
whole is equal to the sum of its parts " involves no 
hypothesis of any kind. 

II. Axioms are Experimental Truths ; In- 
ductions from the evidence of our senses. 

That is, the evidence upon which we believe axioms is 
of the same kind as the evidence upon which we 
believe any other fact of external nature — our ex- 
perience of their truth. They are, in fact, the 
simplest and easiest cases of generalisation from 
the facts furnished to us by our senses or imagina- 
tion. 

(In the following discussion Mill takes, for convenience, the 
particular axiom, "Two straight lines cannot inclose a 
space," or, "Two straight lines, having once intersected, 
continue to diverge, and never again meet." Most of the 
arguments are, however, applicable to any other axiom 
also). 

WheweU's theory of the evidence of axioms, in 
opposition to Mill ; a modified a priori view. 

That axiomatic truths may, indeed, be suggested by 
experience, but it is not experience which proves 
the axiom ; its truth is perceived a priori by the 
constitution of the mind itself, from the first 



DEMONSTRATION. 75 

moment the Proposition is understood, and without 
the necessity of verifying it by trial. " To learn a 
Proposition by experience, and to see it to be neces- 
sarily true, are two altogether different processes of 
thought." 

In answer to this vieiv, and in support of his own, 
Mill first shows that the evidence derived from ex- 
perience is amply sufficient to prove axiomatic truths ; 
and, secondly, examines certain arguments advanced 
against his theory. 

1. The evidence derived from experience is amply 
sufficient to prove axioms. 

Experience confirms them almost every moment of our 
lives ; in the axiom we have taken, for example, we 
cannot look at any two intersecting lines without 
seeing that it is true. Experimental proofs crowd 
in upon us in such profusion, without a solitary in- 
stance of even the suspicion of an exception, that it 
is impossible to conceive more decisive proof from 
experience. Where, then, is the necessity of sup- 
posing any other evidence of the truth of axioms 
than that which is seen to be so amply sufficient ? 

2. The following arguments have been advanced 
against Mills view ; they are given seriatim with 
Mill's replies. 

(a.) If an axiom were proved by the evidence of the 
senses, we could only be convinced of its truth by 
actual trial, as in present case by seeing or feeling 
the straight lines ; but it is seen to be true by 



76 DEMONSTRATION. 

merely thinking of them, therefore the ground of our 
belief must be in the laws of the mind itself. 

Mill replies : — 

Imagination can so perfectly reproduce sensations of 
form, that our mental pictures of lines, circles, &c, 
are just as fit subjects of experiment as the external 
pictures, or the realities themselves. 

(6.) The thing asserted being that the lines will not 
meet in infinity — how can the senses take cognisance 
of a non-existent phenomenon — can we see or feel 
the lines not meet at an infinite distance ? 

Mill replies : — 

We know that if the two lines ever do meet, or even 
begin to approach, this must occur at some finite 
distance ; and the perfection of our mental pictures 
of forms enables us to frame a mental image of the 
appearance of the lines at such a point — an appear- 
ance which would be inconsistent with our notions 
of a straight line. 

(c.) The next and great argument is this : — 

Not only are axioms conceived of as being truths, 
but as being necessary truths — truths which could 
not be otherwise ; Propositions of which the contra- 
dictory is distinctly inconceivable. 

Now, since experience can only inform us of what is, 
has been, or may be, and cannot possibly certify us 
of what must be, axioms cannot be based upon that 
evidence. 

£This argument of Whewell's may be conveniently 

thrown into this technical form : — 
The necessary truth of a Proposition is a mark of its not 



DEMONSTRATION. 77 

being derived from experience (experience cannot 

inform us what must be). 
The inconceivability of the contradictory is the mark of 

the necessary truth of a Proposition ; 
■v The inconceivability of its contradictory is a mark of 

a Proposition not being derived from experience. 
Mill attacks and refutes the minor premiss as below.] 

Mill replies. — Inconceivability of the contradictory 
of a Proposition is so far from being a mark of ite 
(so-called) necessary truth, that it is not even a 
certain mark of its being true at all. Or, What is 
inconceivable is neither necessarily, nor always, 
false ; for — 

1. Inconceivability is an accidental thing, dependent on 

the mental constitution and history of the person 
who tries to frame the conception. Our capacity or 
incapacity of conceiving the truth of a Proposition 
depends chiefly on three things — (1.) The frequency 
and constancy with which we have found the Pro- 
position true; (2.) Whether we have ever found it 
to be false ; and (3.) If not, whether there exist any 
analogies which might suggest the possibility of its 
ever failing to be true. This is further proved and 
expanded in 2 and 3 following. 

2. We have several examples of Propositions, once regarded 

as inconceivable by the greatest men, now recognised 
not merely as conceivable, but as the only true ac- 
counts. 

Thus, to Newton it was inconceivable that a 
body should act where it is not ; yet now it is uni- 
versally recognised in the theory of Gravitation, 
Magnetism, &c. 

3. Conversely we have examples of truths really arrived, 



78 DEMONSTRATION. 

at by long and complex investigations, becoming so 
familiar, that by some scientific men they are held 
to be necessary truths, i.e., truths whose contra- 
dictory is and must always have been inconceiv- 
able. 

Thus, some have supposed the first law of motion to be a 
necessary truth in this sense ; and also the doctrine 
of the uniformity of composition of chemical com- 
pounds. Thus we see that inconceivableness proves 
nothing, except that two ideas are so firmly asso- 
ciated in our minds that we find it impossible to 
disconnect them. 

[Sir Wm. Hamilton coincides with Mill in rejecting incon- 
ceivability as a certain mark of falsity ; to assert, he 
observes, that what is inconceivable is necessarily 
false, brings us into collision with the higher laws 
of thought ; thus, matter must be either infinitely 
divisible or not, in virtue of the " Law of Excluded 
Middle," and it cannot be both of these, in virtue of 
the " Law of Contradiction." Now, either of these 
alternatives is inconceivable. We cannot imagine 
the subdivision of a material particle carried on 
infinitely, nor can we conceive a point at which that 
division must end, an atom so small that it could 
not be divided. But, as just said, by " Law of Ex- 
cluded Middle," one of these inconceivable alterna- 
tives must be true ; therefore something inconceiv- 
able is not necessarily false, q.e.d. 

(By the higher "Laws of Thought" Sir W. H. means 
chiefly these three axioms — " Law of Identity," A 
is A; " Law of Contradiction," A is not non-A ; and 
" Law of Excluded Middle," A is either B or not 

B).] 

£T/oese additional arguments, though not noticed by 
Mill, may be thrown together here, because they are 



DEMONSTRATION. 79 

often given in support of the view that axioms are 
proved by an a priori law of the mind, and not by 
experience. 

1. Increase of certainty, pari passu, with increased ex- 

perience, is a mark of a truth derived from experi- 
ence. For example, after seeing ten people die, I 
should expect more confidently the mortality of any 
new case than if I had seen five only, still more if I 
had seen one hundred than if merely ten, and so on, 
up to full certainty. Axioms want this mark, being 
believed, with the fullest certainty, immediately they 
are understood. 
The reply seems to be — that after having once arrived at 
full certainty, no further experience can increase 
that certainty, and if, as happens in axioms and 
similar simple assertions, a single experience is 
sufficient to fully prove them, no further certainty 
can be given by increased experience, or anything 
else. 

2. Impossibility of establishing a proposition by proposi- 

tions simpler or more certain than itself, is a mark 
of the necessary truth of that proposition. Axioms 
possess this mark, and therefore are necessarily 
true. 

3. Mankind universally, even those who dispute them in 

the abstract, constantly acting as if they believed 
them, is a mark of necessary truth. Axioms pos- 
sess the mark. 

4. There is to our minds a distinct and conscious differ- 

ence between the two classes of truths, both as to 
their certainty (it seems quite impossible to get rid 
of the belief) and the kind of evidence we should 
mentally fall back upon if they were disputed. 
Compare in this respect the proposition, " Two and 
three make five," with the assertion " Fire burns," — 



80 DEMONSTRATION. 

the former seems to have a necessity about it which 
does not belong to the latter. 
The reply is simply — that two and three are just as true 
of propositions which are admitted to be proved by 
experience, and therefore are not marks of (so-called) 
necessary truth ; while four only asserts the in- 
conceivableness of the contradictory in different 
words.] 

III. Herbert Spencer's doctrine of the Uni- 
versal Postulate. 

Mr Herbert Spencer, while agreeing with Mill that the 
inconceivableness of the contradictory of a Proposi- 
tion is not always a mark of its truth, yet main- 
tains the view that this inconceivability of the 
contradictory is really the basis of our belief of 
axioms, because it is the ultimate basis of all our 
beliefs. This is his reason for terming it the uni- 
versal postulate. He lays it down that, whenever 
a Proposition is invariably believed (that is, by all 
men always), it is true ; and the mark of its being 
invariably believed is the inconceivability of its 
contradictory, so that we may phrase the universal 
postulate thus : — 

Whenever the contradictory of a Proposition is incon- 
ceivable, that Proposition must be accepted as 
true. 

This, he says, is really assumed in all our beliefs ; if 
they are intuitive, as when we assert that we taste 
a bitter taste, the real ground of our belief that we 
are experiencing it, is the impossibility of conceiv- 
ing (i.e., believing) the contradictory — that we are 
not. So in belief of inference, we believe that the 
conclusion follows from the premisses, because we 



DEMONSTRATION. 81 

cannot conceive it not following. Being thus the final 
ground of all our beliefs, it must be that of axioms 
amongst the rest. 

This view is unfolded in and supported by these two 
arguments : — 

(1.) Invariableness of belief in a Proposition (of which 
inconceivability of contradictory is the mark) re- 
presents the sum or aggregate of all past experience. 
Facts of every kind are continually coming before 
us, and impressing themselves upon us ; our ex- 
perience is a register of such facts, and the incon- 
ceivableness of a belief shows that it is altogether 
at variance with that register. 

To this Mill replies. — (1.) Even if inconceivableness repre- 
sent the net result of all past experience, why not 
appeal to that experience itself, and not presume 
it from a mere incidental consequence ? (2.) But 
uniform experience is by no means an unfailing crite- 
rion of certain truth ; and (3.) Not only is uniformity 
of past experience far from being a test of certain 
truth, but inconceivableness is very far from being 
a test even of that imperfect test. Uniformity of 
contrary experience is only one out of many causes 
of inconceivableness. 

(2.) "Whether a good proof or a bad one, inconceivableness 
of the contradictory is the best proof we can have, 
since all beliefs are in the last resort founded upon it. 

Mill replies. — That this is not true is proved decisively 
by the facts that some Propositions are believed 
which are actually inconceivable. "We cannot con- 
ceive that a body can act where it is not, that any- 
thing can be created out of nothing, that external 
objects are mere bundles of sensations, and not 
realities, external to us — yet all of these Proposi- 
tions are or have been believed. 



82 FOUNDATIONS OF THE SCIENCE OF NUMBER. 



CHAPTEK VI 
The Foundations of the Science of Number. 

In the previous chapter we have seen that the primary 
truths of Geometry are axioms, and the hypothetical postu- 
lates contained by implication in the definitions. "We have 
seen, moreover, that the axioms of Mathematics (as of other 
branches of knowledge) are arrived at and proved by ex- 
perience, just as Propositions asserting any other law of 
Nature. We next proceed to discuss the Science of Num- 
ber, which includes every branch of Mathematics not in- 
cluded in the Science of Space or Extension (that is, Geometry). 
Practically it consists of Arithmetic and Algebra. 

I. The fundamental Propositions on which 
the Science of Number is based, are: — 



1. Definitions, as 



Two is one and one. 
Three is one and two. 

The sums of equals are equal. 
The differences of equals are equal. 



The definitions, like those of Geometry, define a name 
(" three " means " one and two "), and postulate or 
lime a matter of fact, — that collections of objects 
exist which impress the senses thus .'., or thus . . ., 
and which may be resolved into two others respect- 
ively impressing the senses thus ., and thus . .. 



FOUNDATIONS OF THE SCIENCE OF NUMBER. 83 

II. These fundamental Propositions (i.e., 
the axioms and the postidates in the Definitions) 
of the Science of Number are Inductions, — 
generalisations from experience. 

Nothing need be added to the arguments already given 
under axioms of Geometry ; and we proceed to dis- 
cuss the 

III. Doctrine that the Definitions and 
theorems of the Science of Number are mere 
verbalisms. 

Putting out of view the two axioms, the advocates of 
this doctrine assert that the Propositions of this 
Science are simple transformations in language — 
substitutions of one set of words for another. That 
" three is two and one," say they, is not a state- 
ment of any external fact, but simply a way of 
saying that mankind have agreed to use the word 
"three" as exactly equivalent to "one and two," 
to call by the former name whatever is called by 
the latter more clumsy and less concise name ; and 
so of every other numerical Proposition. 

This vieiv is supported by tivo arguments ; — 

First — We do not carry ideas of any particular 
things along ivith us when ive manipulate 
algebraical or arithmetical symbols (as x or a). 

Mill replies to this : — 

(1.) That an examination of the mental phenomena in- 
volved in numerical processes shows that we have 



84 FOUNDATIONS OF THE SCIENCE OF NUMBER. 

been really dealing with things throughout ; the 
Symbols are things, and our operations upon them 
express facts. For — (a.) These symbols will serve the 
purpose of things ; and (b.) In the processes they 
are treated as things, — i.e., the Propositions we 
make use of therein assert properties of things, and 
not of signs merely. 
(2.) An examination of the results of numerical processes 
will often show us that we have really been dealing 
with things — for the facts at which we arrive in the 
conclusion are often by no means the same as the 
fact or facts from which we started. 

Second — The Propositions of Number, when con- 
sidered as Propositions relating to things, all 
seem to be identical Propositions. 

Thus, "two and one are three" if applied to objects, 
seems to assert not mere equality, but absolute 
identity between the two collections of objects. 

Mill replies — 

It is true that the subject and predicate of a numerical 
Proposition may have the same denotation (i.e., may 
denote precisely the same objects), but they have 
a different connotation (that is, they imply two dif- 
ferent states of those same objects). The Proposi- 
tion given asserts that a collection impressing the 
muscs thus . ., and another thus ., if put together 
will impress senses thus .'. — these several impres- 
sions on the senses are what the names " two," "one," 
and "three" respectively connote; and the Pro- 
position asserting that it the first two collections are 
put together they will impress the senses as in the 



FOUNDATIONS OF THE SCIENCE OF NUMBER. 85 

third, though exceedingly simple and obvious, is yet 
not identical. 

IV. Under ivhat circumstances the postulates 
in the Definitions of Science of Number are 
hypothetical. 

1. The Propositions of pure member (number merely as 

number) are true absolutely without any mixture 
of supposition. Number 3 always = number 1 -f- 
number 2. 

2. But when from equality or inequality of number, 

equality or inequality in any other respect (as 
weight, size, &c), is inferred, then the supposition 
or hypothesis that " all the numbers are numbers 
of the same or equal units" becomes necessary. 
"We cannot be assured that 1 pound -f- 2 pounds = 3 
pounds, unless we suppose 1 pound always to be the 
same. 

V. " The characteristic property, then, of 
Demonstrative Science is that it is hypo- 
thetical" 

By this Mill means that — Demonstrative Science starts 
from the granting of certain fundamental supposi- 
tions, and then proceeds to trace the consequences of 
such assumptions, i.e., what inferences may be drawn 
from them ; leaving for subsequent separate con- 
sideration how far they are true, and what correc- 
tions must be made if they are not exactly true. 

The inquiry, then, as to the inferences which can be 
drawn from assumptions or fundamental Proposi- 
tions taken as settled, is what properly constitutes 
Demonstrative Science. 



86 FOUNDATIONS OF THE SCIENCE OF NUMBER. 

VI. The " Reductio ad absurdum " consists 
in thus assuming a Proposition which we wish 
to prove untrue, and then by inferring from 
it, and deducing an " absurd ;; consequence, 
showing its falsity. 

An " absurd " Proposition here means the contradictory 
of some Proposition which, by the conditions of the 
particular inquiry, is not to be questioned. 

VI I. Some have said — That the ultimate 
"proof of the validity of the Syllogistic process 
is dependent on a " reductio ad absurdum" 

That is, if any one admits the premisses of a Syllogism, 
yet denies the conclusion, by a " reductio ad absur- 
dum" we can compel him to admit two contradictory 
Propositions — that is, one of the premisses and its 
contradictory. If he deny the Syllogism, he can be 
forced to a contradiction in terms. 

Mill remarks — 

This is not so, for since the validity of the Syllogism is 
denied, it is useless to attempt to prove it by a 
process which involves another Syllogism. The 
denier of the Syllogism can only be forced to an 
infringement of the fundamental axiom of ratio- 
cination — "Whatever is a mark," &C. 

VIII. That Proposition, then 9 is logically 



FOUNDATIONS OF THE SCIENCE OF NUMBER. 87 

" necessary" to refuse our assent to which 
would be to violate the above axiom. 

Nothing, therefore, is logically necessary but the con- 
nexion between conclusion and premisses. 

Demonstrative Evidence is that from which anything 
follows by logical necessity, i.e., as conclusion from 
premisses. 



BOOK III. 



INDUCTION. 



CHAPTEK I, 

Preliminary Observations. 

A general Proposition is one in which a Pre- 
dicate is affirmed or denied of an actually or 
potentially indefinite number of individuals, 
viz., all existing or capable of existing in pre- 
sent, past, or future, which possess the attri- 
butes connoted by the subject-name. Or, it is 
One which asserts that one phenomenon 
always accompanies {i.e., is a mark of) another 
phenomenon. 

[We must not, therefore, be misled by the mere verbal 
form of a Proposition. Thus, " All continents 

possess large rivers" is not a true logical general 



PRELIMINARY OBSERVATIONS. 89 

Proposition, but only a bundle of four singular 
Propositions, viz., Europe possesses large rivers, 
Asia, &c., Africa, &c., America, &c. We cannot 
properly say attributes connoted by " continent " 
are marks of attribute " possessing large rivers;" 
the two only happen to be associated in the only 
cases of which we have knowledge, but if a new 
continent were raised, say from the bottom of Pacific 
Ocean, we have no assurance that it would contain 
large rivers. 

On the other hand, " God is a being superior to 
man," is a general Proposition, as much to a Chris- 
tian as to a polytheist, since it means — whenever 
and wherever we meet the attributes connoted by 
" God," there we shall meet attribute " superiority to 
man." 

The distinction between a Proposition really 
general and one only general in its form, will easily 
be made if it be remembered that a* true general 
Proposition asserts that one phenomenon is a mark 
of another phenomenon — and thus such a Proposi- 
tion gives us a power of predicting that when or 
where we meet the former phenomenon we shall 
also meet with the latter. Thus, though we happen 
to be correct in saying, " All the apostles were 
Jews," we cannot predict from this that if an apostle 
were at any future time appointed, he would be a 
Jew, — the attributes connoted by " apostle ,; are not 
marks of the attributes connoted by " Jew."] 

It may be well to draw attention to a slight ambiguity in the 
word "Induction," as used by Mill. Properly it means the 
inductive process or operation itself ; but sometimes the 
result of that process,— the general proposition which is the 
conclusion of the operation. The context will show readily 
in which sense the word is used. 



90 PRELIMINARY OBSERVATIONS. 

Induction is — (a.) an Inference, (&.) estab- 
lishing^ general Proposition, (c.) on the evi- 
dence of particular instances. 

Notice the three clauses, (a.), (&.), (c), since they all are 
necessary to constitute a true Induction. It must 
be an Inference, that is, the conclusion must be 
wider than the total of the premisses ; and we 
say u establishing a general Proposition," because 
although we may argue from particulars to particu- 
lars, yet when we are logically warranted in doing 
this, we may also draw the general Proposition, as 
already explained. This is the sense, then, in which 
the expression must be understood : that in an 
induction we always may draw a general conclusion, 
though, as a matter of fact, we may content our- 
selves with simply inferring to a new particular 
case. 

[Another Definition of Induction: — 

" Induction is that operation of the mind by which we 
infer that what is true in a particular case or cases, 
will be found true in all cases which resemble the 
former in certain assignable respects." 

This is essentially the same, but it is inserted for com : 
parison.] 

Importance of Inductive Logic ; Induction 
the great subject of Logic. 

All Inference (and, therefore, all proof, and all discovery 
of truths not self-evident) consists in Inductions, 
and in the interpretation and application of Indue- 



INDUCTIONS IMPROPERLY SO CALLED. 91 

tions. All our knowledge, therefore, not intuitive 
comes to us fundamentally from the same source. 

The Logic of the Sciences is, therefore, the 
Logic of every-day life. 

The same principles and processes apply to the infer- 
ences we are continually making in common affairs, 
and to the establishment of Scientific Principles. 



CHAPTEE II. 

Inductions improperly so called. 

L Inductions improperly so called are : — 

1. Mere Verbal Transformations. 

2. Mathematical ll Inductions 7 ' \ Nearl [ o "J" 3 " 

W (a.) Proving geometrical 

6. Inferences by Farity I \ theory by a diagram. 
of Reasoning, as — ( ) ^ rillin s u p terms of a 

J \ series. 

4. Colligation of Facts. 

1. Mere Verbal Transformations. 

That is, when we affirm of a class simply what has 
already been laid down as true of each and every 
individual in it separately ; so that the conclusion 



92 INDUCTIONS IMPROPERLY SO CALLED. 

is only a summing up and reassertion {i.e., " verbal 
transformation ") of the premisses. Thus, Europe 
has large rivers, so Asia, so Africa, so America ; 
•\ All continents have large rivers. Paul was a 
Jew, Peter was a Jew (and so on through the whole 
list) ; ■■« All the apostles were Jews. 

There is here no Induction, for — (1.) There is no infer- 
ence; nothing more is stated in the conclusion than 
in the premisses ; and (2.) The seemingly general 
Proposition is only a number of single Propositions 
written in a compendious and abridged form. 

It may be added, this species of false Induction is the 
only form of Induction recognised by ordinary 
Logicians, who term it " True Logical Induction." 

2. Mathematical " Inductions." 

This process is of this kind : After having proved separ- 
ately the following Propositions — 
(1.) A straight line cannot cut a circle in more than 

two points. 
(2.) A straight line cannot cut an ellipse in more than 
two points. 
Similarly of the parabola and hyperbola — 
Conclude that 

" A straight line cannot cut a conic section in more 
than two points." 
The conclusion in such cases is a really general Proposi- 
tion, but the process is not Induction, for there is no 
inference; nothing but a summing up of the pre- 
misses, as in the last case. 

3. Inference by Parity of Reasoning. 

That is, when, though the conclusion arrived at is really 
general, yet we do not believe it on the evidence of I 



INDUCTIONS IMPROPERLY SO CALLED. 93 

the particular case or cases themselves, but because 
we see that the same evidence which established 
the particular cases, will also prove every other case 
' coming under our conclusion. Amongst examples 
of this are : — 

(a.) Proving a geometrical theorem by means of a diagram. 

Thus having proved that the three angles of the triangle 
ABC are equal to two right angles, we conclude that this 
is true of every triangle, not because we find it true of 
ABC, but for the same reasons which proved it true of 
ABC. 
(6.) Filling up the terms of a series when the law of the series 
has been ascertained. 

4. Colligation of phenomena or descriptions. 

Colligation of phenomena is the forming a general 
motion or conception of those phenomena, such gene- 
ral notion being constituted of the common attri- 
butes or properties of the phenomena colligated. 
This notion (the sense of the agreements of the 
phenomena) being expressed in words, constitutes 
a description of them. Thus, suppose we contem- 
plate the whole animal creation, and discovering the 
points of agreement in, or the attributes common 
to every member thereof, we may by combining 
those points of agreement frame a general notion 
"animal," which notion unfolded in words would 
give a description of animal, — a notion and a 
description which would apply to every member of 
that department of creation. 

Mill defines Colligation thus ; — 

Colligation is that mental operation which enables us 
to bring a number of actually observed phenomena 



94 INDUCTIONS IMPROPERLY SO CALLED. 

under a general description ; or which enables 
us to sum up a number of details in a single Pro- 
position. 
There are two questions at issue between Mill and 
Whewell with respect to Colligation : — 

1. As to the exact nature of Colligation, or the pro- 

cess of forming general notions. 

2. The relation between Induction and Colligation. 
The first is discussed in Book iv., chapter 2 ; the second 

we now proceed to examine. 

II. Relation between Induction and Col- 
ligation. 

According to Whewell — " Induction is the Colligation of 
phenomena by means of appropriate conceptions, — 
in short, Colligation is Induction." 

Mill replies. — The two processes are quite distinct ; 
Colligation, or the formation of a general conception 
of the phenomena to be investigated, is a necessary 
preliminary to Induction; but Induction is some- 
thing more than Colligation ; for — 

1. There is no real inference in mere Colligation. 

2. Successive Colligations, though conflicting, may yet 

all be correct as far as they go ; that is, they may 
all correctly represent the facts observed at the 
time they were respectively framed. Now, it 
would be absurd to assert that conflicting Induc- 
tions could all be true. Thus, the successive 
notions which the ancients formed of the nature 
of the paths of the heavenly bodies were all correct 
as (Icscri/jf/o/is, i.e., as far as they represented the 
facts known at the time. Suppose 1 observe certain 
facts, and attempt to describe them to another, 
it is evident that my description is correct, as a 



INDUCTIONS IMPROPERLY SO CALLED. 95 

description, if I succeed in conveying to him the 
same notion as he would have had of the facts if 
he himself had observed them in my stead. But 
if new facts are discovered, a new, and perhaps 
different, notion or description will be required, 
which again may have to give place to a third, 
differing from both, if new facts in due course 
come to light ; yet every one of these notions and 
descriptions will have been in its turn correct as 
a description of the facts known at that time. 
3. Colligation only describes; Induction, besides in- 
cidentally describing, also predicts and explains. 
Whewell, in fact, confounds conception, — the process 
of forming general notions (or, as he calls it, " Col- 
ligation") with Induction. When we conclude by 
an Induction that " All men are mortal," he would 
represent the process as consisting simply and solely 
in framing a general notion of man, which general 
notion should include " mortality," and then com- 
paring this general notion with observed facts to 
see if it agrees therewith ; and if not, framing another 
and different notion, comparing it in like manner ; 
and so on till we find a notion which will correspond 
to the facts. It may at first sight seem strange 
that two such apparently distinct processes as In- 
duction and Conception should be thus confounded ; 
but compare Bain's "Senses and Intellect" under 
" Similarity " (Eeasoning and Science). 



96 GROUND OF ALL INDUCTION. 

CHAPTEK III. 

Ground of all Induction. 

I. Fundamental axiom or ground of In- 
duction. 

There is an assumption implied in every case of Induc- 
tion, and which assumption is found to be true as a 
matter of fact : — 

Loosely expressed — This fundamental assumption is, that 
the course of Nature is uniform — as far, at least, as 
regards the phenomena we are concerned with in 
the particular Induction. 

More accurately — That what is true in certain cases is 
true of every other case resembling the former in 
certain assignable respects. 

II. How this assumption is involved in any 
Induction. 

An Induction may be thrown into the form of a Syllo- 
gism by supplying a major premiss, thus : — 

Whatever is true of A, B, C, is true of all men, 
A, B, C are mortal, 
.•. All men are mortal. 

Now, it is evident that this major premiss is nothing 
else but an assertion of the uniformity of nature 
in so far as regards the phenomena we are concerned 
with. " Uniformity in nature " means that what 






I 



GROUND OF ALL INDUCTION. 97 

we find in one or more cases we shall continue to 
find in similar cases, which is exactly our major 
premiss. 
But this major premiss resembles the major premisses 
of any other Syllogism ; that is, it is no part of the 
evidence which proves the conclusion, but only a 
mark that there is sufficient evidence to prove the 
conclusion, so that if false the conclusion is falla- 
cious. In fact, as already explained, in connexion 
with the theory of the Syllogism, both the conclu- 
sion and the major premiss are alike conclusions 
from the antecedent observed particular cases. We 
see A, B, C, and everybody else in whose case the 
experiment has been fairly tried, die, — this is our 
evidence, and from it we conclude "All men are 
mortal ; " but it is that very same evidence that gives 
us our assurance that " what, in this respect, is true 
of A, B, C, is true of all men," in other words, our 
observed cases prove to us that nature is uniform 
in respect of the connexion between humanity and 
mortality. 

III. How uniformity of Nature is proved. 

This ultimate principle is a generalisation from all our 
Inductions, it is a conclusion by an Induction, "per 
enumerationem simplicem" from a large number of 
Inductions. 

For example, suppose we knew nothing of the 
principle, i.e., did not know whether Nature was 
uniform or not, yet in the Induction in II., we could 
conclude that Nature was uniform in respect, at 
least, of the connexion between men and mortality ; 
so also in a second Induction we might prove Nature 
uniform in another respect ; in a third, and so on. 

G 



98 GROUND OF ALL INDUCTION. 

As fresh instances of proved uniformity were added 
to our list, we might begin to suspect that Nature 
was always uniform ; new cases of Induction con- 
stantly being made, each in its own sphere proving 
Nature uniform, would proportionally strengthen 
that suspicion ; and when finally age after age 
passes away, and Inductions innumerable are made, 
every one of which adds its item of proof without a 
single contradictory instance (that is, an instance of 
Nature capriciously varying being found), the in- 
ference of the universality of the principle is irre- 
sistible. Before proceeding farther it is necessary 
to explain :— 

Induction, "per Enumerationem Simplicem" 
is thus defined by Bacon. 

" Obi non reperitur instantia contradictoria," i.e., Induc- 
tion, because we have never found an instance to 
the contrary. It is an argument from simple, 
unanalysed experience ; its formula being, " Such 
and such has always been found to be true, no 
instance to the contrary has ever been met with ; 
therefore such and such is true." All crows hitherto 
observed have been black, no crow of any other 
colour has ever been seen, therefore all crows are 
black. 

Mill's definition is, the ascribing the character of a 
general truth to any proposition which happens 
to be true in every instance we have known of, 
— i.e., to which wo do not happen to know any 
exception. 

It is the sort of Induction natural to untutored minds, 
and is usually, but not always, very fallacious. The 
following is the 



GROUND OF ALL INDUCTION. 99 

Condition which renders an Induction per 
Enumerationem Simplicem a valid process. 

We must know that if any exception ever had occurred 
we should be aware of it ; in other words, precisely 
in proportion as its subject matter is limited and 
special, so is the process unreliable. 

This necessary assurance we cannot in the great majority 
of instances obtain ; yet it is the fact that there 
do exist certain remarkable cases where, having 
this certainty, an Induction by simple enumeration 
amounts to a rigorous proof, indeed the only proof 
of which these cases are susceptible. These are 
(1.) Fundamental principles of Mathematics ; and 
(2.) The principle of the Uniformity of Nature. 

The axiom of the uniformity of Nature, then, is proved 
by this form of Induction ; the evidence consisting 
in this, that the principle has been found true in 
every legitimate Induction hitherto made, and never 
once false ; while, at the same time, from the fact 
that these innumerable Inductions cover the whole 
field of Nature's operations, we are entitled to con- 
clude that any real exception must have come under 
our notice. 

IV. The chief merit of Bacon as regards 
Inductive Philosophy lay in his pointing out 
the insufficiency of this loose and merely pas- 
sive mode of Induction, and the essential 
importance of an active interrogation of Nature 
by experiment. 



100 LAWS OF NATURE. 

CHAPTER IV. 

Laws of Nature. 

I. The general regularity in Nature is an 
aggregate of particular uniformities called 
Laws. 

(" Uniformity''' — A uniform conjunction of phenomena, 
either by way of coexistence, the two phenomena 
always being found together, or by way of sequence, 
one phenomena always being followed by the other.) 

That the general uniformity of Nature is made up of 
uniformities in particular respects requires no illus- 
tration, being self evident. 

II. Laws of Nature. 

(1.) In the loose sense. — A law of Nature is a proposi- 
tion expressive of any sufficiently well ascertained 
uniformity. 
(2.) In a stricter sense. — A law of Nature is any estab- 
lished uniformity which cannot be accounted for 
by, or resolved into, simpler uniformities. The 
expression " Law of Nature " is commonly employed 
with a tacit reference to the true signification of 
" law," the expression of the will of a superior ; and 
hence is employed to designate such uniformities 
only as might be regarded as separate expressions 
of the creative will ; and is not usually applied to 
such uniformities as can be shown to be mere 



LAWS OF NATURE. 101 

results of such fundamental laws. Thus, that " the 
mercury will rise in an upright exhausted tube, 
whose orifice is immersed a little beneath its sur- 
face," is a proved uniformity, which, however, may 
be resolved into two more fundamental principles 
— gravity, and the uniform transmission of pressure 
by a liquid ; and as these two cannot be resolved 
into more fundamental principles, they rank at 
present as laws of Nature, while the derivative law 
or uniformity does not. 

III. Scientific Induction must be grounded 
on previous spontaneous Inductions. 

Spontaneous Inductions are those which are so palpable 
as to be made without conscious effort, — induc- 
tive inferences which force themselves upon men's 
minds, — as that " fire burns," " water quenches 
thirst," and so on. Now such Inductions as these 
give us that insight into the order of Nature which 
is necessary before we can lay down the principles 
of Induction ; if a rational being were suddenly 
created and dropped upon our earth, however great 
his intellectual powers, it would be utterly impos- 
sible for him to frame inductive canons ; he would 
not know whether caprice or uniformity prevailed, and 
if uniformity, under what circumstances it might be 
expected to be manifested. There is nothing im- 
possible in the supposition that the arrangement 
might be such that what has the properties of iron 
at one moment might have those of ice the next ; 
that, in a word, uniformity might be replaced by 
the wildest caprice, or that the uniformity, even if 
found, might differ widely from that which we 
actually experience. In fact, we require experience 



102 LAWS OF NATURE. 

to show us how far and under what conditions experi- 
ence is to be relied upon. 

IV. The stronger (i.e., more certain) Induc- 
tions are the tests to which we endeavour to 
bring the iveaker. 

Suppose, for instance, that we possess a strong Induction 
to this effect, — " every effect must have a cause," 
it is evident that if by any means we can bring a 
weaker generalisation within this better established 
law, — i.e., if we can show that either the weaker 
generalisation must be true or our strong Induction 
must be false, — the weaker is at once raised to the 
same degree of certainty with the stronger. 

Since, then, the logical method of proving a gene- 
ralisation is thus to bring it within a more certain 
generalisation, the inquiry necessarily arises — 

Do any great Inductions exist, thus fitted to be 
ultimate tests of all others f 

There are such Inductions, certain and universal, and it 
is because there are such that a Logic of Induction 
is possible. 

The universal Law of Causation is such an Induc- 
tion, and the four Inductive methods of Mill are 
simply expedients for bringing weaker generalisa- 
tions, as far as possible, under this great law, — that 
is, methods of proving that such generalisations are 
deducible from it. 






LAW OF UNIVERSAL CAUSATION. 103 

CHAPTEE V. 
Law of Universal Causation. 

I. Law of Universal Causation. 

Every phenomenon which has a beginning must have a 
cause ; and it will invariably arise whenever that 
certain combination of positive facts which con- 
stitutes the cause exists, provided certain other 
positive facts do not exist also. 
This law contains two clauses : — 

(1.) That every phenomenon which has a beginning 

must have some cause. 
(2.) Given the cause the effect will invariably follow, 

provided that counteracting causes do not exist. 

II. Definition of Cause. 

(1.) If we regard cause as including all the antecedents, 
both positive and negative (" positive " = what must 
be present ; " negative "== what must be absent). 

A cause is that assemblage of phenomena, which 
occurring, some other event follows, invariably and 
unconditionally. 

(2.) If we restrict the meaning of cause to the assemblage 
of positive conditions : — 

A cause is that assemblage of phenomena, which 
occurring, some other event follows invariably, sub- 
ject only to negative conditions. 

" Unconditionally" in this definition, means under any 
imaginable supposition as regards other things. 



104 LAW OF UNIVERSAL CAUSATION. 

This is what some express by saying that an effect 
follows its cause " necessarily." To be a cause, it is 
not enough that the sequence is invariable ; night is 
an invariable sequence of day, but day is not the 
cause of night. The sequence must be unconditional 
also, — given day, night should follow, whatever we 
choose to suppose about other things (as, for in- 
stance, that rotation of earth should cease), to justify 
us in calling day the cause of night. 
The negative conditions of an effect may be summed up 
in this, — absence of counteracting or preventing 
causes, 
(a.) Counteracting causes. — Most causes counteract the 
effects of other causes by the operation of the 
very same law as that by which they produce 
their own effects. Each law is fulfilled, — each 
cause in reality has its effect, but the effects 
neutralise each other more or less. Thus, if a 
spout supply a cistern at a certain rate, while a 
precisely similar spout empties it simultane- 
ously, the effect, the filling of the cistern, is 
defeated, though the cause, the influx of water, 
cannot be spoken of otherwise than as really 
producing its own proper effect. 
(6.) Preventing causes. — Some causes seem, however, to 
be simply preventive, — i.e., destroying an effect, 
(not by producing their own, but) by simply ar- 
resting it. Opacity is a phenomenon of this kind. 

III. Popular distinction between the " con- 
ditions " and the " cause " of an effect unphilo- 
sophical. 

Strictly speaking, the cause is the sum total of all the 
conditions or circumstances necessary for the pro- 



LAW OF UNIVERSAL CAUSATION. 105 

duction of the effect — the aggregate of the ante- 
cedents thereof. In popular language, however, it 
is usual to single out one of these antecedents as 
the cause, the remainder being termed conditions of 
the effect. Thus, suppose a stone to be dropped 
into water and allowed to sink, — the sinking of the 
stone is an effect, the antecedents being (1) the 
mutual attraction between stone and earth ; (2) the 
stone being within the range of that attraction ; (3) 
the specific gravity of the stone being greater than 
that of the water ; and, finally, the negative condi- 
tion, absence of support for the stone. Now, in 
common discourse, any one of these conditions, — 
the entire sum or aggregate of which is properly the 
cause, — may be called the cause of the sinking of 
the stone. Even a mere negative condition (i.e., 
the absence of something) is often spoken of as if 
the positive cause of an effect (as, the absence of 
the sentinel was the cause of the army being sur- 
prised), whereas it is evident that no mere negation 
can produce an effect, but can only not hinder its 
being produced. 

The distinction thus popularly made is usually based 
upon either : — 

(1.) That one of the antecedents which comes last, 
and is thus an event completing the sum of 
conditions which forms the cause, and upon 
which the effect immediately follows, is termed 
the cause. 

(2.) That one of the antecedents which is most pecu- 
liar and special to the aggregate of antecedents, 
is often popularly the cause. 

(3.) So also that one of the antecedents which is least 
likely to be known to the hearer. 



106 LAW OF UNIVERSAL CAUSATION. 

IV. The Universal Law of successive pheno- 
mena is the Law of Causation, 

(This does not mean that two successive phenomena 
are necessarily cause and effect, — day and night, 
for instance, are not ; but any phenomenon which 
succeeds another must be a phenomenon having a 
beginning, and therefore a cause, and consequently 
must come under the law of Causation.) 

Phenomena in nature may stand to each other in two 
relations, — that of Simultaneousness {Co-existence in 
Time), or that of Succession. The most valuable 
truths with which we have to do are truths of Suc- 
cession : on our knowledge of these depends all our 
power of foreseeing future facts, and of influencing 
these facts for our own benefit. Hence we see why 
the main business of Inductive Logic is with cases 
of Causation — with determining what are the effects 
of given causes, and what the causes of given effects. 

[V. Distinction between " agent " and 
" patient " merely verbal. 

Many make a distinction between the thing acting (the 
agent) and the thing acted upon (the patient). Thus, 
if a man be poisoned by prussic acid, the poison 
would be reckoned as an " agent," the nervous sys- 
tem of the individual as the " patient." 

This distinction is, however, merely in language ; 
"patients" are always "agents" which have been 
implied in the words describing the effect. It is 
evident, for instance, that the peculiar properties 
of the nervous system are as much involved in, are 
as much conditions of, the death as the properties 
of prussic acid.] 



LAW OF UNIVERSAL CAUSATION. 107 

VI. "Permanent Causes'' or "Primitive 
Natural Agents " include all substances and 
phenomena which do not begin to exist, — i.e., 
which might, for aught we can see in them to 
the contrary, have existed from all eternity, 
and may never have had a beginning at all. 

Such may he either : — 

(1.) Objects — the sixty-three or more elementary bodies, 
with their various properties and the combinations 
of such found in nature, — the atmosphere, water, 
&c. ; the heavenly bodies, sun, moon, and stars. 

(2.) Events — i.e., cycles of events — periodical cycles being 
the only form in which an event can have perman- 
ence. Such are the rotatory and orbital motions of 
the earth, of the moon, &c. 

Of these original Causes it may be observed : — 

(1.) We cannot tell why any one of them exists at all ; 
we can give no account of their origin. 

(2.) We cannot tell why they exist in a particular man- 
ner — why one is found in one place, another in 
another. 

(3.) No law of their mode of distribution can be dis- 
covered. 

(4.) Every phenomenon which has a beginning, every 
caused phenomenon, — i.e., every phenomenon except 
the primeval causes themselves, — must arise or have 
arisen immediately or remotely from one of these 
primeval causes, or from some combination thereof. 



108 ON THE CONJUNCTION OF CAUSES. 

CHAPTER VI 

On the Conjunction of Causes. 

I. When two or more Causes act together 
so as to intermix or combine their effects, one 
of two things may happen, either : — ■ 

1. The joint effect is of the same kind with the separate 

effects ; the laws work together without alteration. 
In this case we may speak of the mixed effect as 
consisting of the separate effects. This constitutes 
" Composition of Causes" and such an effect is termed 
a " Compound Effect" 

For example, suppose a force acting on a particle 
in the direction of the north, and another tending 
to pull it to the west, the two forces are two con- 
joined causes, and the effect, which is a force lying 
along diagonal of their parallelogram, is of the same 
kind with the separate effect of each cause. The 
effect of each of the two forces is, in fact, found in 
the conjoined effect of both. This is, therefore, a 
case of " Composition of Causes," and the effect is 
"Compound." 

2. The joint effect is not of the same kind with the 

separate effects ; the separate effects of the causes 
disappear, and a totally new set is developed by 
their combination. Here we may speak of the con- 
joint effect as being generated or produced by the 
.simple effects. This is the case of " Combination of 
( W//.XV.S-." aiid such an effect is a " Jleteropathic Effect." 
For example, the substance iodine manifests, or 



ON THE CONJUNCTION OF CAUSES. 109 

is a cause of, certain properties or effects — dark 
colour, peculiar smell, metallic taste, volatility. So 
also potassium causes us to feel a metallic lustre, 
&c. But if these two causes, iodine and potassium, 
be united, we find hardly any trace of the effects 
which they produced when separate ; those effects 
indeed are gone, and are replaced after the union of 
the causes by a totally distinct set — white crystal- 
line structure, solubility in water, &c. So, again, 
the properties of a mixture of oxygen and hydrogen 
are chiefly the sum of those of the separate gases 
conjoined — a compound effect ; but let the mixture 
be exploded, the two causes, oxygen and hydrogen, 
combine, but no trace of the properties of the 
separate gases can be found in water. This effect, 
therefore, is heteropathic. 

II. Composition of Causes. 

1. The causes may act in the same or in different direc- 

tions, — two forces may pull a body along the same, 
or along intermediate, or along opposite lines. Two 
spouts may either both fill, or one may empty and 
the other fill, a cistern. Causes may, in fact, appear 
to annihilate each other's effects, — apparently pro- 
ducing no effect at all, — as when two equal forces 
act in opposite directions upon a particle, yet each 
cause really exerts its full efficacy according to its 
own law. 

2. The Composition of Causes is the more frequent case, 

not only absolutely in the case of simple causes, but 
for the reasons given below in III. and IV. 

III. The total effect of conjoined Causes may 



110 ON THE CONJUNCTION OF CAUSES. 

be partly compounded, partly heteropathic, and, 
In fact, is never wholly heteropathic. 

There are no cases of causation, from conjoined causes, 
in which the resultant phenomena do not in some 
respects obey the principle of the Composition of 
Causes. The weight of a combination of iodine and 
potassium, for instance, is always the sum of the 
weights of the separate ingredients. 

IV. Heteropathic phenomena, when they act 
together, may compound their effects. 

Laws generated by combination may act with one another 
on the principle of composition. As a single example, 
iodide of potassium and of water — both generated 
heteropathically — may, if mixed, give the sum or 
aggregate of their separate effects, as regards taste, 
chemical reactions, &c. 

V. That effects are proportional to their 
Causes — 

Is, when true, a case of the composition of causes, a cause 
then being compounded with itself. If, for example, 
a column of mercury be heated through 1°, a certain 
expansion follows, through 2° twice as much, and 
so on, the effect being so far proportional to the 
cause. It is clear, however, that in this case, the 
total effect of a rise through several degrees is com- 
pounded of the separate effects of a rise through 
each degree. If, however, the cause be augmented 
beyond a certain limit, the proportionality ceases, 
the mercury being converted into vapour, — a hetero- 



OBSERVATION AND EXPERIMENT. Ill 

pathic effect. So we shall always find that the 
axiom holds good as long as every fresh increment 
of the cause compounds its effect with that of the 
preceding parts of the cause, but that it fails the 
moment the composition of causes fails, and the 
effect becomes heteropathic. 



CHAPTEK VII. 

Observation and Experiment. 

I. The order of Nature, looked at as a whole ; 
presents a vast mass of causes followed by a 
vast mass of effects, and, therefore, Inductive 
inquiry, — having for its object the ascertaining 
what causes are connected with what effects, 
and what effects with what causes, — is in some 
sort a process of Analysis, and presents three 
steps — (1.) The mental separation ; (2.) The 
actual separation; and (3.) Varying the cir- 
cumstances. 

The first step in Inductive inquiry is a 
mental analysis of the complex whole presented 
by Nature in the aggregate into its several 
parts. 



112 OBSERVATION AND EXPERIMENT. 

The whole order of nature, as perceived at the first 
glance, consists of a great mass of phenomena, fol- 
lowed in the next instant by another great mass of 
phenomena. The first step, then, consists in learn- 
ing to see in the aggregate antecedent a number of 
separate antecedents, and in the aggregate conse- 
quent a number of separate consequents. 

The second step is an actual separation of 
the elements of complex 'phenomena. 

We must obtain some of the antecedents apart to be able 
to try what will follow from them, or some of the 
effects apart to find out by what they were preceded. 

The third step, determining what antecedents 
and what consequents are connected, 

Is accomplished by varying the circumstances, — that is, 
by obtaining instances of the phenomenon we are 
investigating, which, by differing in some of their 
circumstances, will throw light on the inquiry. 

II. For the purpose of varying the circum- 
stances, — that is, of obtaining a number of dif- 
ferent kinds of instances of the phenomenon 
in question, we must have recourse to either 
observation or experiment. That is, we must 
either find suitable instances in Nature, or we 
must make them by an artificial arrangement. 

It is highly important, however, to remember that the 
logical value of an instance is entirely dependent 



OBSERVATION AND EXPERIMENT. 113 

upon what it is in itself, and not upon the mode in 
which it is obtained, yet there are important prac- 
tical distinctions between observation and experi- 
ment, which it is necessary to notice. 

III. Experiment is our resource when we 
wish to determine the effect of a given cause, 
for we can take a cause and try what it will 
produce, but we cannot take an effect and try 
what it was produced by 

Advantages of Experiment are ; — • 

(1.) It enables us to multiply our instances indefinitely, 

(2.) To isolate the phenomenon we are studying. 

(3.) To vary the surrounding circumstances indefinitely, 
and thus, amongst other things, 

(4.) To obtain the precise sort of instances we require. 

(5.) To these may be added, that, since our only way to 
prove that one thing is the cause of another, is to 
take the supposed cause and try whether it will pro- 
duce the effect, and since we cannot generally do 
this by observation, experiment is usually our only 
means to prove causation. 

IV. Observation is chiefly applicable when 
we are unable to obtain artificial experiments, — 
that is, when we are investigating the unknown 
causes of a given effect. 

Suppose we wish to ascertain the unknown causes of a 
given effect, cholera ; now, since we know not its 

11 



114 OBSERVATION AND EXPERIMENT. 

causes, we cannot produce cholera artificially, and 
our only resource is to wait till nature produces 
instances for us, by observation of which we may 
hope to discover by what they have been invariably 
preceded. 
Observation alone, without aid from experiment, can rarely 
prove cause and effect. It may show us that two 
phenomena are invariably conjoined, that where we 
find one we shall also find the other, but it will not 
go beyond this ; we cannot be sure that the two 
phenomena are cause and effect : they may, for ex- 
ample, be both effects of some common cause. To 
prove cause and effect, we must take the cause and 
try what effects it produces, — a matter of experi- 
ment, as already said. 

V. From this contrast between observation 
and experiment, an important conclusion 
follows : — 

That in the sciences in which artificial experiment is im- 
possible or very limited, direct Induction is practised 
at a disadvantage, amounting generally to imprac- 
ticability. 

The methods of such science must therefore be chiefly 
Deductive. 



THE INDUCTIVE METHODS. 115 

CHAPTEKS VIII and IX. 

The Inductive Methods. 

[Preliminary Remarks. 

1. Mill speaks of these as the " Four Methods of 
Experimental Inquiry," and these are : — 

1. The Method of Agreement and the Joint Method. 

2. Of Difference. 

3. Of Residues. 

4. Of Concomitant Variations. 

" The Joint Method" is not reckoned separately, inas- 
much as it is really an employment of the two forms 
of the method of agreement together, — the Positive 
and the Negative. It is termed by Mill either the 
"Joint Method of Agreement and Difference" (or 
shortly, "The Joint Method"), or "The indirect 
Method of Difference." 

2. When these are spoken of as Methods of Ex- 
perimental Inquiry, the term experimental must be 
taken as equivalent to experiential, — Methods of 
Inference from experience generally, and not merely 
from experiment in the strict sense. The Method 
of Agreement, for example, usually derives its in- 
stances from observation. 

3. The Method of Agreement and of Difference 



116 



METHOD OF AGREEMENT. 



are the two fundamental Methods ; the others being 
only special forms of one of these, or of both together. 
Thus: — 



Method of agreement, 
(where we find cause 
we find the effect, 
and vice versa,) 



Includes also the Joint 
Method (in part — the nega- 
tive partaking of Method 
of Difference). 



2. Method of Difference,] Indudes algo Method of 
(where one is absent I ResidueSj Method of Con „ 

orremoved, the other ., . Tr • ,. 

' I comitant V ariations. 

is also.) J 

It is easy to see how the Method of Concomitant Varia- 
tions partakes of the nature of the Method of Dif- 
ference ; two phenomena (A and a) are found con- 
joined, and the (partial) removal or (partial) adding 
on of one is followed by a corresponding change in 
the other, which is essentially the Method of Dif- 
ference. 

4. In the exposition of the Inductive Methods, 
Mill takes the simplest possible case, — that is, he 
supposes every effect. 

(1.) To always have exclusively one and the same cause, 
and (2.) To be always distinct from — not in any way 
intermixed with — any other coexistent effect. In 
what way it becomes necessary to modify our pro- 
ceedings when we come to the practical use of the 
methods is discussed in Chapter X.] 



I. — Method of Agreement. 

Canon. — If two or more instances of the presence 
of the phenomenon in question have only, in common, 



METHOD OF AGREEMENT. 117 

the presence of one other circumstance, that circum- 
stance, in the presence of which alone all the in- 
stances agree, is the cause or effect of the given 
phenomenon. 

Its principle is — that of comparing different cases 
in which the given phenomenon occurs, in order to 
discover in the presence of what these instances 
agree. 

Exemplification. 

Suppose the given phenomenon is cholera, and we wish 
to ascertain its cause : by this method we should 
have to compare a number of instances of cholera, 
to determine by what it had invariably been pre- 
ceded. It is clear that the cause of cholera must 
be amongst these invariable antecedents ; and if we 
can be sure that in each case we know everything 
which has preceded (or all the antecedents of) the 
attack, and if } in a number of cases, only one circum- 
stance can be found which has invariably preceded, 
that one must be the cause of cholera. 

Remarks : — 

1. The possibility of the plurality of causes introduces the 
possibility that the two phenomena, thus apparently 
connected, may only be conjoined casually or by 
chance. This is called by Mill "the characteristic 
imperfection" of the Method of Agreement. 

Most effects may be produced by a plurality of 
causes, — i.e., by different causes in different cases ; 
thus, the phenomenon "death" may be caused in 
one case by disease, in another by poison, and in a 



118 METHOD OF AGREEMENT. 

third by injury. This possibility does not, however, 
radically vitiate the method, but only renders it 
necessary that its first results should be corrected 
by the process for the " Elimination of Chance " 
(which see). 

2. There is, however, another imperfection in this 

method, which prevents us from ever proving by 
it more than that two phenomena are invariably 
conjoined ; it cannot demonstrate that they are 
cause and effect. This, which may be termed " the 
practical imperfection " of the method, is the impos- 
sibility of assuring ourselves that we know all the 
antecedents in our instances. 

To take our previous example, we might find that 
the drinking of a certain sort of impure water was 
alone an invariable antecedent, as far as we can see, 
of cholera. Suppose, further, we had assured our- 
selves that these two phenomena were not merely 
accidentally found together in our instances, still 
we could not be assured that they were cause and 
effect. The impure water and the cholera attack 
might both be results of some obscure cause which 
had wholly escaped our observation — some unknown 
atmospheric state, for example, forming an ante- 
cedent in our instances, of which we were ignorant. 
The possibility of the presence of unknown antecedents, 
then, is the reason why the Method of Agreement 
can only yield empirical laws, and cannot prove 
causation, and is, therefore, chiefly useful as afford- 
ing suggestions, or as an inferior resource where 
better methods are impracticable, — i.e., where arti- 
ficial experiments cannot be made. 

3. Since this method is our chief resource when we wish 

to determine the causes of a given effect, it mostly 
makes use of instances obtained by observation. 



METHOD OF AGREEMENT. 119 

4. This method does not require instances of a very 
special and definite kind ; any instances whatever, 
in which the phenomenon occurs, may be examined 
for the purposes of this method. 

Negative Method of Agreement, 

The Canon is the same as that of the Positive 
form of the method, with the substitution of the 
word " absence" for " presence/' wherever it occurs. 

The principle is that of comparing different in- 
stances of the absence of a phenomenon, to discover, 
if possible, in the absence of what other thing alone 
those instances agree. In other words, to fix upon 
the one thing alone which is never found where our 
phenomenon is absent. 

The negative Method of Agreement is not affected by 
the possibility of a plurality of causes, but it is 
affected by the possibility of the absence of unknown 
antecedents, just as the positive form is by the pos- 
sibility of their presence. 

The further consideration of this form of the Method of 
Agreement is deferred till we come to speak of the 
" Joint Method ; " for though it is true abstractedly 
that it might be worked alone without the positive 
method, yet practically it is quite impossible to 
make much use of it by itself. The difficulty of 
showing that the instances agree, in the absence of 
one thing only, is almost insuperable. But when 
the positive method has suggested a cause, we can 
then inquire, with some chance of success, whether 
that is not the only thing universally absent when 
the effect is absent. 



120 METHOD OF DIFFERENCE. 



II. — Method of Difference. 

Canon. — If an instance in which the phenomenon 
in question occurs, and an instance in which it does 
not occur, have every other circumstance in common 
save one, that one occurring only in the former, — 
that circumstance in which alone the two instances 
differ, is the effect or the cause, or a necessary part 
of the cause, of the phenomenon. 

The principle is that of comparing an instance of 
the occurrence of a phenomenon with a similar in- 
stance in which it does not occur, to discover in what 
they differ. 

Remarks : — 

1. This method is more particularly a method of artificial 

experiment (its ordinary use being to compare the 
condition of things before, with those after, an ex- 
periment), because — 

2. It is commonly employed to determine the effects of 

given causes ; and because — 

3. The instances which it requires are rigid and definite 

— they must be exactly alike, except that in one the 
phenomenon must be present and in the other 
absent. 

4. If this method is inapplicable, it is usually because 

artificial experiment is impracticable. 

5. It is the only method, of direct experience, by which 

laws of causation can be proved. 

6. If the instances fulfil exactly the requirements of the 

Canon, this method is perfectly rigorous in its proof. 



THE JOINT METHOD. 121 

Many of our inferences in daily life are simple applica- 
tions of the Method of Difference. 

Thus, a man in full life receives a shot in his 
heart and becomes a man dead. We infer that the 
wound caused death, because it is the only circum- 
stance in which the case in which death is found 
differs from the case in which death is not found. 



III. — The Joint Method. 

Canon. — If two or more instances of the presence 
of the phenomenon in question have only in common 
the presence of one other circumstance ; while two or 
more instances of the absence of the phenomenon 
have in common the absence of that circumstance 
only, — that circumstance in which alone the two sets 
of instances differ, is the effect or the cause, or a 
necessary part of the cause of the phenomenon. 

Remarks : — 

1. The Joint Method is really a double employment of 

the Method of Agreement, thus : 

"We observe a number of instances in which the 
phenomenon is present, and find them to agree only 
in the presence of a given circumstance. (Positive 
form.) 

We observe a number of instances in which the 
phenomenon is absent, and we find that that same 
circumstance is the only thing which is uniformly 
absent. (Negative form.) 

2. It is called, also, " The Indirect Method of Difference" 

because the negative instance is got not by direct 



122 METHOD OF RESIDUES. 

experiment, but indirectly, by showing what would 
be the result if experiment could be made. 

3. The Method of Difference compares two instances ; 

the Joint Method compares two sets of instances. 
The proof derived from one set is independent of 
that derived from the other, and corroborative of it. 
Still both together do not amount to a proof by the 
direct Method of Difference, on account of the pos- 
sibility of the presence or of the absence of un- 
known antecedents in the positive and negative sets 
respectively. 

4. The Joint Method is, however, a great extension of 

the Simple or Positive Method of Agreement ; hav- 
ing this great advantage over it, of not being affected 
by the possibility of the plurality of causes. 



IV. — Method of Kesidues. 

Canon. — Subtract from any phenomenon such part 
as is already known to be the effect of certain ante- 
cedents, then the residue of the phenomenon is the 
effect of the remaining antecedents. 

Exemplification. — Suppose we have several phe- 
nomena ABC followed by several others a b c, and 
that we know A to be the cause of a, and B of b, 
then C must be the cause of c. 

Remarks :—- 

1. This method is, in fact, a modification of the Method 
of Difference; but the negative instance {i.e., where 
phenomenon is absent) is obtained by Deduction, 
not by direct experience. The Deduction being 



METHOD OF CONCOMITANT VARIATIONS. 123 

this, — from the known effects of A and B separately, 
we infer their effect conjointly, and subtract this 
effect from the total effect, ah c. 

2. This method would be equally rigorous with Method 

of Difference, if (1.) we could be certain of the total 
effects of the known antecedents (A and B), and (2.) 
that the remaining antecedent (C) is the only one 
present. 

3. This being generally impracticable, we must complete 

the evidence derived from this method, either — 
(1.) By applying Method of Difference, — i.e., ob- 
taining supposed cause (G) separately, and 
trying its effect ; or 
(2.) By the Deductive Method, — i.e., we must 
account for C's agency when suggested, and 
infer it deductively from established laws. 

4. This method is the most fertile in unexpected results. 

Y. — Method of Concomitant Variations. 

Canon. — Whatever phenomenon varies in any 
manner whenever another phenomenon varies in 
some particular manner, is either a cause or an effect 
of that phenomenon, or is connected with it by some 
link of causation. 

The principle is — that even if we cannot remove 
an antecedent altogether, yet we may be able to 
modify it in some way short of its total removal. 

Its Axiom is — Anything upon whose change, the 
change of an effect is invariably consequent, must be 
the cause or be connected with the cause of that 
effect. 



124 METHOD OF CONCOMITANT VARIATION. 

Remarks : — 

1. The changes or variations with which this method is 

chiefly concerned are either — (1.) Changes in quan- 
tity, or (2.) Changes of position in space. 

2. To logically infer causation through this method, we 

must first determine that the variations in the two 
phenomena are really concomitant. This is proved 
by the Method of Difference, — that is, we retain all 
the other antecedents unchanged, while the particu- 
lar one is subjected to the requisite variations. 

3. This method may usefully follow the Method of 

Difference, to determine according to what law the 
quantities or relations of the effect follow those of 
the cause. 

4. The most striking application of this method is to 

cases where we have to determine the effects of 
those of the Permanent Causes, which we cannot 
wholly remove. 

Such causes are — the earth, with its gravitative, 
magnetic, and other properties ; the sun, moon, and 
stars, with any known or unknown properties they 
possess. These causes we can never wholly remove, 
but we can modify them, — we can get nearer to or 
farther from the centre of the earth, we can remove 
a magnetic needle from one place to another, and 
by such methods, varying the supposed cause, and 
noticing the consequent variations of the effect, we 
are able to determine the effects of these irremov- 
able permanent causes, and separate them from the 
effects of other causal agencies. 

5. The most satisfactory application of this method is in 

cases where the variations in the cause are variations 
in its quantity. 

This kind of variation in a cause is generally ac- 



GENERAL REMARKS. 125 

companied not only by variation in the effect, but 
by composition of causes, with a proportional varia- 
tion thereof. 
We may have two cases of this relation of variation : — 
(1.) Where cause and effect vanish together. 
(2.) Where they do not vanish together, — i.e., where 
one is reduced to 0, the other has still some 
positive value. 
Two precautions are necessary in drawing conclusions 
from this kind of concomitant variation : — 
(1.) We ought to be able to determine the absolute 
quantities of the antecedent (A) and consequent 
(a). 

For if we do not know the absolute quanti- 
ties of A and a, we cannot tell the exact 
numerical relation according to which these 
quantities vary. We cannot say that we have 
twice, three times, &c., as much of a thing, un- 
less we know the quantity of once the thing. 
(2.) We must remember that the law which the quan- 
tities seem to follow within the limits of our 
observation may not hold beyond those limits. 



VI. — General Kemarks. 

1. These four Inductive Methods are the 
only possible modes of inquiry by experience, 
or a posteriori. These, therefore, with such 
assistance as can be obtained from Deduction, 
constitute the available resources of the human 
mind for determining the Laws of the Succes- 
sion of Phenomena. 






126 GENERAL REMARKS. 

2. Whewell makes the following objections 
to these Methods : — 

(1.) They assume the very thing which is most 
difficult to obtain, — the reduction of an argument to 
a formula. 

Mill replies : — 

This objection is exactly analogous to that brought 
against the Syllogism, by those who said that the 
great difficulty is to get your Syllogism, not to judge 
of it when obtained. As a matter of fact, both of 
these objections are true, but still the canons and 
formulae fulfil their logical function, that is, they 
enable us to judge of evidence when found, — the 
very office of the Science of Logic, and by no 
means a superfluous one, as the commonness of 
false inferences testifies. 

(2.) No discoveries have ever been made by their 
means. 

Mill replies : — 

This objection proves too much, for since the methods 
are formulae of the only possible modes of inference 
from experience, this assertion is equivalent to 
saying that no discoveries were ever made by 
experience. 



PLURALITY OF CAUSES. 127 



CHAPTEK X. 

Part I. — Plurality of Causes. 

I. By Plurality of Causes is meant simply 
this, — that a given effect may arise from dif- 
ferent causes in different cases ; thus, the 
phenomenon " death " may be caused in one 
case by disease, in another by violence, in a 
third or fourth by poison or old age. 

II. The possibility that the effect we are in- 
vestigating may have a plurality of Causes, 
leads to " the characteristic imperfection " of 
the Method of Agreement (in its positive form 
at least). 

Thus, suppose we have a group of causes, ABC, followed 
by a group of effects, a b c, and ABE by ad e (the 
fact being that A is the cause of a, B of b, and so 
on, though we are not supposed to know this) ; by 
the Method of Agreement we conclude that A is 
the cause of a, — since B or C cannot be, because 
absent in second instance, nor B nor E because 
absent in first ; but the moment we admit that a 
might have a plurality of causes, this conclusion 
fails, — it might be, then, produced by B or C in the 
first, and by B or E in the second case. 



128 PLURALITY OF CAUSES. 

III. How to correct the uncertainty arising 
from this cause. 

The Method of Agreement is not radically vitiated 
by this imperfection ; for the two phenomena found 
together must either (1.) have no connexion, — i.e., be 
conjoined by chance, or (2.) must have some con- 
nexion. 

(1.) By merely multiplying instances of the same kind, 
we shall get data for determining whether the coin- 
cidences are more frequent than chance will account 
for ; if so, we may conclude there is some connexion 
(between our A and a). 
(2.) If, however, we sufficiently multiply and vary our 
instances {i.e., get them as the canon prescribes, 
agreeing only in one other circumstance (A) besides 
a, the phenomenon in question), having ABC fol- 
lowed by a b c, A D Ehy a cle, A F G by afg, &c, 
we may be sure either : — 
(a.) That a has as many causes as there are instances, 
— that is, that A and a are only conjoined by 
chance. Remedied as in (1). 
(b.) That A and a are joint effects of some unknown 

antecedent existing in all our instances ; or 
(c.) That A is the cause of a. 
We can never wholly get rid of the possibility of the 
presence of an unknown antecedent, and all there- 
fore we can conclude by this method is that A and 
a are found together, — where A is we may expect a, 
or vice v&rsd. 

| IV. We may recapitulate here the use of 



I 



PLUKALITY OF CAUSES. 129 

mere number of instances of the same 
hind : — 

1. They enable us to have proportionate assurance that 

no error has been committed in the observation of 
the particular facts. 

2. They furnish data for eliminating chance, — i.e., for 

showing that the conjunctions between two pheno- 
mena are more frequent than mere chance will 
account for. 
When these two objects have been fully attained, 
nothing whatever is added to the certainty of our 
conclusion by mere repetition of similar instances.] 

V. To determine Causes of a given effect, 
producable by a plurality of Causes. This is 
done either : — 

(1.) By separate inductive inquiries, each cause being 
tested by a separate series of investigations ; or, 

(2.) By collecting a number of instances of the occur- 
rence of the effect, and finding that while they agree 
in no one antecedent, yet they always agree in the 
presence of one out of a certain number thereof. 
Thus, the effect " death" is always preceded by dis- 
ease, old age, poison, or violence. 

VI. Plurality of Causes does not affect the 
Negative Method of Agreement nor the Method 
of Difference. 

1. In negative method we show that instances in which 
a is absent agree only in not containing A. Now, if 

I 



130 INTERMIXTURE OF EFFECTS. 

this be so, A must not only be the cause of a, but 
the only possible cause. For if a is absent, all its 
causes must be absent in every case, but the only 
thing uniformly absent in our instances is A, .". A 
is the only possible cause of a. 
2. The Method of Difference is obviously not affected. 
If we have two cases, — A B C followed by a b c, and 
B C by b c (i.e., A being taken from ABC gives B C 
followed by b c, or being added to B C followed by b c, 
gives A B C by a b c), it is certain A is the cause of 
a in that case, whatever other causes of a may exist 
in other cases. 



Part II. — Intermixture of Effects. 

1. We have already explained what is meant 
by a complex or intermixed effect (see Chap, 
vi., Book iii.). 

It is an effect resulting from the conjunction of several 
causes. Thus, if a person suffering from severe ill- 
ness were to take some poison, his death might be a 
complex effect, resulting partly from disease, partly 
from poison ; so in the parallelogram of forces or 
velocities, the movement of the particle along the 
diagonal is a complex effect — the intermixture of 
the effects of two causes, viz., the two forces tend- 
ing to carry it along the two sides of the parallelo- 
gram. 

We have .seen also that such complex effects are divisible 
into two distinct classes— (1.) Compound Effects, 
where the separate effect of each of the causes really 
continues to be produced, and these separate effects 



INTERMIXTURE OF EFFECTS. 131 

unite into one aggregate or sum — the complex effect : 
and (2.) TIeteropathic Effects, where the separate effect 
of each cause ceases entirely, — a perfectly different 
phenomenon resulting from the conjunction of the 
causes. 

There is, however, a special form of Heteropathic Effects, 
which require to be separately noticed — Transfor- 
mations, where cause and effect are mutually conver- 
tible — £.&, where we can make A produce a, or a pro- 
duce A. Thus, hydrogen and oxygen, when fired, 
produce water ; water, galvanized, produces hydro- 
gen and oxygen. In this case the problem of find- 
ing a cause resolves itself into the much easier one 
of finding an effect, a problem to the solution of 
which experiment, and, therefore, direct Induction, 
is especially applicable. 

With the exception of " Transformations" the investiga- 
tion of complex effects by direct Induction is prac- 
tised at such great disadvantages as generally to be 
impracticable. Our resource in such cases is the 
Deductive Method. This is especially true of the 
first division of complex effects, viz., — Compound, 
Effects ; and in them, this inapplicability of direct 
Induction (observation and experiment) is, cost, par., 
in direct proportion to the number of the causes 
which conjointly produce the intermixed or com- 
plex effect, and to the smallness of the share 
which any one of these causes has in producing that 
effect. Mill proceeds to prove this in detail (see II.) 

II. The investigation of a complex effect 
may he conducted either : — 

1. Deductively — by computing a priori what would be the 
effect of the conjoint causes. 



132 INTERMIXTURE OF EFFECTS. 

2. By Induction, either by way of : — 

(a) Simple observation — simply collecting instances of 

the effect as they occur ; 

(b) Experiment — making instances — taking the sup- 

posed causes and trying what effects they produce 
when conjoined. 

Now Mill goes on to show that neither by (a) nor 
(b) can we effect much in investigating complex 
effects. 

1. Method of simple observation inapplicable. 

Take this example — "recovery from consumption" — a 
complex effect ; is the " taking cod-liver oil " one of 
its causes ? 

It is obvious that many separate causes must combine 
to produce our effect ; now in such a case, where 
many causes are acting to one end, the share of each 
cause in the effect is not in general very great, and 
hence the effect is not likely to follow very closely 
any single cause in its presence or absence, still less 
in its variations. 

2. Method of Experiment is inapplicable, because 
we are tenable to take certain precautions necessary 
to the scientific employment of experiment. These 
are : — 

(a) No unknown circumstances must exist in our cases. 

For instance, we should know everything (which can 
influence consumption) which exists in the system, 
when \vc give the oil. 

(b) Known circumstances must not have effects liable to 

be confounded with the effect of the cause we are 
studying. 



INTERMIXTURE OF EFFECTS. 133 

The most, then, we can hope to obtain by direct Induc- 
tion in complex effects is, that a given cause is very 
often followed by a given effect. 

To sum up then : — 

Of complex effects, Transformations are the best adapted 
to inquiry by direct Induction ; next, the remainder 
of Heteropathic Effects ; and least of all Compound 
Effects, in proportion as the conjoined causes are 
numerous, and as each has but small share in pro- 
ducing the total complex effect. In all such cases 
of the inapplicability of Induction, the Deductive 
Method is our grand resource. 

III. Laws of Causation must he expressed 
as tendencies only. 

Every law of causation is liable to be counteracted, and 
apparently frustrated, by coming into contact with 
other laws, the results of which are more or less op- 
posed to its result. Hence, with many such laws, 
cases in which they are entirely fulfilled do not, at 
first sight, seem instances of their operation at all. 
Suppose a ball to receive simultaneously in two ex- 
actly opposite directions two equal impulses, either 
of which would carry it a hundred feet in its own 
direction, the ball would, of course, remain unmoved, 
and no result would appear to follow, yet, in reality, 
each force fulfils its own law, and the ball occupies 
the same position as it would have done if the forces 
had acted successively instead of simultaneously. 
We must, therefore, define force as that which tends 
to cause motion in a body ; and so in every other 
case of causation ; for though a cause always tends to 
produce its effect, counteracting causes may prevent 
that effect being manifested in the usual form. 



134 



THE DEDUCTIVE METHOD. 



IV. There is no such thing as a real excep- 
tion to a general truth. 

The notion that there may be arises from neglecting the 
proper mode of expressing a law, as just explained. 
What is called an exception to a general principle is 
always a case of some other law interfering with it, 
and disguising or destroying its effect. 



CHAPTEE XI. 
The Deductive Method. 

I. The Deductive Method considers separately 
the causes which enter into the Complex: Effect, 
and computes or calculates that effect, d priori, 
from the balance or product of the effects of 
the different causes which produce it. 

Its problem, in fact, is to find the law of a 
Complex Effect from the laws of the different 
causes of which it is the joint result. 

To take a simple example: — Suppose a particle at A is 
r, simultaneously acted upon by 
two forces, one of which, acting 
alone, would earry it to B in 
one second, and the other, in 
like manner, would carry it to 
i) 0. Here the two forces are 
two causes uniting to produce a complex effect — 




THE DEDUCTIVE METHOD. 135 

motion along the diagonal to D in one second. If, 
however, we suppose ourselves ignorant of what the 
result would be, it is evident we might discover it 
either by making numerous experiments with two 
forces, which would always give us a similar motion 
along the diagonal, — this would be direct Induction ; 
or we might calculate a priori what the j oint effect 
of the two causes or forces must be : we might argue, 
for instance, that the particle must evidently travel 
a distance equal to A B to the right of A, and a dis- 
tance of A C below A ; and the point D is the only 
point which fulfils these conditions, ■■■ the particle 
at the end of one second must be at D, and so on ; 
— this would be to apply the Deductive Method. 

II. The Deductive Method consists of three 
distinct operations or steps : — 

1. Ascertaining the laws of the separate causes 

by direct Induction. 

2. Eatiocination from the Simple laws to the 

Complex case, — i.e., calculating from the 
laws of the causes, what effect a given 
combination of them must produce. 

3. Verification by specific experience. 

1. It is first necessary to ascertain the laws of the 
separate causes. 

This is generally done by direct Induction, but if any of 
the separate laws be themselves complex, they may 
have been obtained by a previous Deduction. But 
even then, such complex laws being ultimately de- 
rived from simple or elementary laws (which are 



136 THE DEDUCTIVE METHOD. 

always established by direct Induction), must ulti- 
mately be based upon inductions. 

To this Induction it is essential : — 

(1.) To know what the causes actually are whose effects 

we are about to study. 
(2.) Their laws must then be ascertained. 

This can only ultimately be done by the four Inductive 
Methods. And since the accuracy of this Induction 
is the foundation of the whole inquiry, it is neces- 
sary that, (a) if possible, we must study each of the 
concurrent causes in a separate state ; for (b) if this 
is not possible — if we cannot try the effect of each 
cause apart from others, as in Biology — we experi- 
ment under great disadvantages. 

2. Ratiocination is the second step, — i.e., calculat- 
ing from the knoivn laws of the separate causes what 
effect any given combination of them will produce. 

On this we may observe : — 

(1.) When our knowledge of the laws of the causes ex- 
tends to the exact numerical relations which they 
observe in producing their effects, the ratiocination 
or calculation may reckon amongst its premisses all 
the theorems of the science of Number. Thus, we 
have a planet in its orbit round the sun at any mo- 
ment under the influence of two separate causes — 
the central force of gravity pulling it towards the 
sun, and the tangential force. The actual path 
which it traces is the complex or joint effect of 
these two causes. Now it is clear that if we not 
only know the laws of these separate causes, but are 
able to state them with numerical exactness (as that 






THE DEDUCTIVE METHOD. 137 

the central force varies inversely as the square of 
the distance and directly as the masses), any of the 
properties of squares, square-roots, &c;, which we 
previously know, may be available in the calculation 
of the actual orbit. 

(2.) When the effect takes place in space, and involves 
motion and extension, the theorems of geometry as 
well as of number come in as premisses. This is the 
case, for example, in Mechanics, Optics, Acoustics, 
and Astronomy. In the parallelogram of forces or 
velocities we are able to make use of any of the pro- 
perties of parallelograms, triangles, &c, which we 
think fit in our calculation. 

It might here, however, be naturally asked — How are we 
to be assured of the correctness of our calculation '] 
How can we know that we have taken all the causes 
of our complex effect into account and rightly com- 
puted their joint results ? To this we reply that we 
cannot have this necessary assurance of complete 
accuracy until we apply the proper test — verification 
by experience. Without this, deductive calculation 
is often nothing more than guesswork. 

3. Verification by comparison with the results of 
actual experience is the third step in an inquiry by 
the Deductive Method. 

Our calculations having led us to conclude that the effect 
will be of a certain kind, we must determine by 
actual observation or experiment whether it really 
is so or not. Having, from a knowledge of the laws 
of the central and tangential forces, calculated that 
a planet will move in an ellipse, we must verify the 
result by observing whether its successive places are 
really points on such a curve. 



138 THE DEDUCTIVE METHOD. 

Two particular cases or forms of Verification 
may be noticed. 

(1.) When the theory thus derived leads deductively to 
previously known empirical laws of the phenomena 
in question. 

Thus Kepler's three laws were known as empirical 
laws — i.e., as results of actual observation — before 
the time of Newton, who showed, however, that they 
were deducible from, i.e., were results of, his theory 
of gravitation. 

This is the most effectual verification possible. 
(2.) When the theory is found to be in accordance with 
a complex or obscure case. 

Thus the general law — " Heat is developed by 
compression of air" — was found to explain the ob- 
served fact that the calculated velocity of sound was 
less than the actual velocity. Now though we could 
hardly have discovered the law in question from this 
complex and obscure manifestation of it, yet when 
it was found that it exactly accounted for the differ- 
ence observed, an important verification was sup- 
plied thereby. 

[III. The Deductive Method presents several 
forms, according (1.) To the subject matter to 
which it is applied ; and (2.) According to the 
mode of its application. 

1. (a.) The Abstract Deductive Method, which deals with 

the laws of those sciences which are not concerned 
with causation, and therefore which arc not liable 
to counteraction, — the laws of number and exten- 



EXPLANATION OF LAWS OF NATURE. 139 

sion for example. Euclid's Geometry is an instance 
of the abstract or geometrical method. 
(b.) Concrete Deductive Method deals with those 
sciences which are concerned with phenomena of 
causation. 

2. (c.) Direct Deductive Method — in which we obtain our 
conclusion or law by Deduction first (i.e., by a cal- 
culation of the effects of the conjoint causes), and 
afterwards verify by comparison with the results of 
experience. 
(d.) Inverse Deductive Method — in which we obtain 
our law more or less conjecturally by direct experi- 
ence, and afterwards verify it by showing that it is 
deducible from more general or better known laws. 

In the Direct Method we compare the result of a calcu- 
lation with experience ; in the Inverse we compare 
experience with the result of a calculation.] 



CHAPTERS XII and XIII. 
Explanation of Laws of Nature. 

I. Definition of Explanation. 

An individual fact is said to be explained 
by pointing out its cause, — that is, by stating 
the law or laws of causation of which its pro- 
duction is an instance. 

A Law or uniformity is said to be explained 
when another law (or laws) is pointed out, of 



140 EXPLANATION OF LAWS OF NATURE. 

which that law is itself but a result, and from 
which it may be deductively inferred. 

[Popular and Philosophic Explanation. 

We must remember that all laws of nature are equally 
mysterious ; we can no more assign a why for the 
more general than for the more special laws. But 
popularly an explanation means the substitution of 
a mystery which has become familiar, and so ceased 
to seem mysterious, for one to which we are still 
unaccustomed. An explanation in the philosophi- 
cal sense, meaning merely the resolution of a law 
into more general laws, often does precisely the 
reverse of this, — it resolves a phenomenon with 
which we are familiar into one of which we pre- 
viously knew little or nothing ; as, for instance 
when the familiar law, " All bodies tend to fall to 
the earth," was subsumed into or found to be a case 
of the previously unknown law — " Every particle of 
matter attracts every other."] 

II. Explanation of Laws may take place in 
one of these three modes : — 

1. Besolving the law of a compound effect into 

the separate laws of the concurrent causes, 
and the fact of the coexistence of those 
causes. 

2. Detecting an intermediate link in the 

sequence of causation. 

3. The subsumption of less general laws into a 

more general. 



EXPLANATION OF LAWS OP NATURE. 141 

1. The first is the case of the composition of causes, pro- 

ducing a joint effect equal to the sum of the sepa- 
rate effects. The explanation of such an effect evi- 
dently involves two things: — (1.) the simpler laws 
of the separate causes ; and (2.) the fact of the co- 
existence of those causes (for if not coexistent they 
could not intermix their effects). 

Thus, in explaining the Compound Effect, — the 
orbit of a planet, — we must not only show that and 
how it results from the laws of the simpler causes, 
gravity and the tangential force, but also that those 
causes are actually conjoined, do really act on the 
planet. 

In this case the one law is resolved into two or 
more laws, all of which are more general and more 
certain than that law. (See p. 140.) 

2. The second mode of explaining a law is to point out an 

intermediate link between an effect and its assigned 
cause, to show that this assigned cause is only the 
cause of the cause. A is supposed to cause C, but 
it is found that B is really the cause of C, and A is 
only the cause of B. 

In this case, too, the one law (A causes C) is re- 
solved into two or more laws (A is cause of B, B is 
cause of C), each more general and more certain 
than the original law. 

3. The third mode is the subsumption of less general 

laws into a more general one ; that is, the less gene- 
ral laws are found to be merely instances of the 
operation of the more general. 

Thus the law that bodies fall to the earth was 
found to be a case of the great law of gravitation — 
the laws of magnetism a case of the laws of electric 
currents, &c. 

In this case two or more laws are resolved into 



142 EXPLANATION OF LAWS OF NATURE. 

one, which law is evidently more general than the 
laws gathered up into it ; but as to certainty no 
difference exists, since the less general laws are in 
fact the very same as the more general, and any ex- 
ception to them would be an exception to it also. 

III. Laws are always resolved into laws 
more general than themselves. 

A law is said to be more general than another law when 
it extends to all the cases which that other extends 
to, and to others in addition. 

This is self-evident in the third case. In the first 
and second, we find that the concurrence of two or 
more laws is required to give the less general law ; 
thus, the law " A is followed by B" and the law 
" B is followed by C" are clearly more general than 
" A is followed by G r ," because, for instance, "i is 
followed by i?" is fulfilled not only when B also 
produces C, but also in all other cases where the 
tendency of B to produce C is in any way counter- 
acted. And, besides in the first case, the less gene- 
ral is fulfilled only in the cases where the simpler 
laws are coexistent in the required manner, while 
separately these simpler laws are fulfilled in many 
cases where the condition is wanting, in addition. 

IV. In the first and second cases a law is 
resolved into laws which are more certain than 
itself ; in the third there is no difference in 
this respect. 

Laws are said to be certain in proportion as they are 
liable to exception, — i.e., less liable to bei 



LIMITS TO EXPLANATION OF LAWS OF NATURE. 143 

counteracted. It is perfectly clear that where a 
law is compounded of several others, the chance 
of its being counteracted is very much greater than 
that of any of the simpler laws which compose. 
Each of these separately has only its own chances 
of counteraction, but the complex law has the sum 
of the chances of all. The chances of failure some- 
where in a chain is very much greater than the 
chance of failure in any one particular link. 



CHAPTEK XIV. 

Limits to Explanation of Laws of Nature. 

I. We may recognise two kinds of Laws or 
Uniformities in Nature : — 

1. Ultimate Laws. 

2. Derivative Laws. 

Ultimate Laws are those which cannot be resolved into 
(or deduced from or explained by) other and more 
general laws in any of the three modes of explanation 
just noticed. 

This must be understood in a sense similar to that in 
which chemists speak of an "element," i.e., some- 
thing which cannot, by any known means, be re- 
solved into simpler constituents. 

Derivative Laws — those which can be thus resolved into 
other and more general laws. 



144 LIMITS TO EXPLANATION OF LAWS OF NATURE. 

Now, out of the total number of supposed Ultimate Laws, 
Science is continually removing some by reducing 
them to the class of Derivative Laws, — that is, 
showing that they are mere results of wider prin- 
ciples ; and it becomes an interesting question how 
far this process may go on — to what extent may we 
expect to reduce the number of real Ultimate Laws ; 
what indication have we as to the probable number 
of fundamental and Ultimate Laws, which being 
given, all other uniformities in Nature would follow. 



II. The Ultimate Laws of Nature cannot 
possibly be less numerous than the distinguish- 
able sensations or other feelings of our nature, 
— that is, those feelings which are distinguish- 
able in hind or quality, and not merely in 
degree. 



- 



To explain — I am conscious of a certain sensation called 
a sensation of red, and also at times I am conscious 
of a sensation called a sensation of sound. Now these 
being phenomena which have a beginning, must have 
each its immediate cause, some antecedent which is 
invariably and unconditionally followed by a sensa- 
tion of red, and some other followed similarly by a 
sensation of sound. Call the former cause A, and 
the latter B ; now sound being a sensation different 
in kind from red, the law in virtue of which A is 
followed by a sensation of red, must always be dis- 
tinct from the law by which 11 is followed by the 
mental state known as a sensation, or feeling, of 
sound. The one law can never be resolved into 
the other, they must always remain distinct Ulti- 



i 



LIMITS TO EXPLANATION OF LAWS OF NATURE. 145 

mate Laws ; and so of every other case of sensations 
distinct in kind, — each must have its own ultimate 
law. From this it follows that — 

III. The ideal limit of the explanation of 
natural phenomena would be to show that 
each distinguishable variety of our feelings has 
only one sort of cause. 

That is, to show that, whenever, for instance, I am con- 
scious of a red colour, that particular kind of sensa- 
tion has always the same immediate cause, or ante- 
cedent, that our A is the same in every instance ; 
and so of each distinct sensation. 

IV. In what cases, then, has Science been 
most successful in explaining phenomena, — 
that is, in showing that supposed Ultimate 
Laws are really derivative % 

Chiefly in the case of motion, for these reasons: — (1.) 
That phenomenon is always the same to our sensa- 
tions in every respect, except as regards degree (for 
in the case where there is the greatest semblance of 
difference, motion in a straight line and curvilinear 
motion, the latter is only motion continually chang- 
ing its direction) ; and (2.) it is a phenomenon which 
has an immense plurality of remote causes, — me- 
chanical force, chemical, vital, electrical action, &c. 

Now there is no absurdity in supposing that in 
every case motion may have the same immediate 
cause, or antecedent ; and if such be the case, we 
may expect to bring the cases of motion produced 

K 



146 ON HYPOTHESES. 

by some of these remote causes, under the same 
principle which operates when others of these re- 
mote causes give rise to motion. Accordingly the 
greatest achievements of science have consisted in 
doing this, as when the law of the fall of heavy 
bodies to the earth was found to come under the 
principle of gravitation, when magnetic movements 
were resolved into those produced by voltaic cur- 
rents, &c. 



CHAPTEK XI V— (continued) . 

On Hypotheses. 

I. A hypothesis is any supposition which we 
make, with avowedly insufficient evidence, in 
order to endeavour to deduce from it conclu- 
sions in accordance with facts which are known 
to be real. 

II. The purpose for which Hypotheses are 
framed is either : — 

1. The discovery by anticipation of a law of nature, — 
the hope being that the hypothesis is a correct 
statement of the real law. Now the only way to 
assure ourselves of this is to make inferences from 
the hypothesis, and by comparison of the results 



ON HYPOTHESES. 147 

with actual facts to prove or disprove our supposi- 
tion. It is for this reason that Mill lays down that 
a "legitimate" or "genuinely scientific hypothesis'''' 
must be a verifiable hypothesis, — one, in its own 
nature, capable of being proved or disproved, — one 
destined not always to remain a hypothesis, but 
either be converted into a proved law of nature or 
abandoned as an error. 

2. To fulfil certain subordinate but indispensable func- 
tions, — chiefly (a) to suggest new lines of investiga- 
tion, and (6) to enable us to link together and form 
and retain a clear conception of facts already known. 

It is not meant that any hypothesis, perhaps, is framed 
for either purpose exclusively, but that in most one 
or the other predominates. Many suppositions 
which can never perhaps be proved or disproved, as 
the atomic theory and the electrical theory of mag- 
netism, have been in the highest degree serviceable 
as furnishing suggestions, and a clear order for the 
facts known. 

III. There are two classes of Hypotheses : — 

In the^rs^ the cause (if it be a case of causa- 
tion) is real, but the law of its action is as- 
sumed ; in the second we assume a cause which 
is supposed to act according to known laws. 

1. Hypotheses of the first class — forms of: — 

(a.) Where, in a case of causation, we assume a law for a 
known actual cause, — i.e., a cause not merely exist- 
ing in nature somewhere, but known to have some 
actual influence on the given effect ; or at least to 



148 ON HYPOTHESES. 

be one of a limited number, some of which are 
known to have such influence. 

(b.) Where the hypothesis relates not to causation at all, 
but to the law of correspondence between facts 
which accompany each other in their variations. 

For example, the hypotheses as to the law of the 
variation of the inclination of the refracted ray as 
the incident ray varies its angle of incidence, before 
the true law was known. 

(c.) Hypothetical descriptions — that is, all suppositional 
modes of merely describing phenomena. 

Thus, when we speak of " sun-rising," " setting," 
&c, these are merely suppositional modes of de- 
scribing the phenomena visible. " It is as if so and 
so were the case" is the formula, and all that is 
necessary in any particular case is that this state- 
ment should be true. 

In all these cases, verification is proof ; the hypothesis 
may be received as true, merely because it explains 
the phenomena, since any hypothesis different from 
the true must lead to false results. 

2. Hypotheses of the second class include those 
in which we assume a cause of whose connexion 
with the given effect we are not certain, or even, 
perhaps, of its actual existence in Nature at all. 

In such cases we cannot have the assurance that a false 
law cannot lead to true results. 

IV. Conditions under which a hypothesis 
may be received as true : — 

]. We may dismiss those of the second class by remark- 
ing that the condition in them is that they should 



ON HYPOTHESES. 149 

be reducible to the first. It is indispensable that 
the actual existence of the assumed cause, and its 
connexion with the effect, should be capable of being 
proved, and by evidence other than that of the facts 
which it is adduced to explain. Such a cause was 
what Newton meant by a vera causa. We say other 
and independent evidence, because a hypothesis of 
this class cannot be received as true merely because 
it explains all the known phenomena, for where we 
are at liberty to feign a cause, there is hardly any 
limit to the possible suppositions which will do this. 
Dr Whewell is wrong in laying down that such a 
hypothesis is to be received as true merely because 
it explains the phenomena already known, or even 
because its anticipations turn out to correspond 
with fact. 
2. Of a hypothesis of first-class the conditions are : — 
(a.) It must lead deductively to true results. 
(b.) The case must be such that a false law can- 
not possibly give the true results. 

Both of these are included in this one canon — 
that the final step, the verification shall amount to, 
and fulfil the conditions of, a rigid Induction. 

Such an Induction falls into the formula of the 
Method of Difference. 

V. Subordinate functions of hypotheses. 

It must not be assumed from what has been said that it- 
is never allowable to imagine a cause ; all that has 
been laid down is that such a supposition must not 
be received as true, merely because it explains the 
phenomena. The subordinate functions, — that of 
suggesting new lines of inquiry, and of affording a 
clear and cornected view of known facts, and which 



150 ON PROGRESSIVE EFFECTS. 

are absolutely indispensable, are often as effectually 
fulfilled as by hypotheses which we put forth as 
demonstrably representing the true law or fact. 

VI. Some inquiries which deal with fore- 
gone collocations of causes are not hypotheti- 
cal but inductive. 

There is a great difference between inventing laws of 
nature to account for phenomena and merely endea- 
vouring to conjecture what collocation now gone by 
may, in conformity with known laws, have given 
rise to facts now in existence. The latter is the 
strictly legitimate operation of inferring from an 
observed effect, the existence in time past of causes 
similar to those by which we know the effect always 
now to be produced. 

Thus, we have before us a certain effect, say the 
arrangement of certain geological strata ; we know 
what causes and collocations would produce such an 
arrangement now, and from this we endeavour to 
infer what causes and collocations might have really 
formerly produced the effect in question. 



CHAPTER XV. 
On Progressive Effects. 

I. A Progressive Effect is a complex effect, 
arising from the operation of one cause, by the 
continual addition of an effect to itself. 



ON PROGRESSIVE EFFECTS. 151 

Thus, the fall of heavy bodies to the earth, sixteen feet 
in first second, forty-eight in second, and so, from 
action of one cause, gravity ; continuous rusting 
of iron exposed to moist air, are progressive 
effects. 

II. There is an obvious distinction between 
temporary and permanent effects. 

Temporary, like a flash of lightning or ex- 
plosion of gunpowder ; permanent, those effects 
which remain unless some cause interfere to 
alter or destroy them. 

Now, an agent or cause producing a permanent effect 
may, instead of being merely temporary, be itself 
permanent. In this case whatever effect has been 
produced up to a given time would subsist perma- 
nently (absence of altering causes being supposed), 
even if the cause were then to perish. Since, how- 
ever, the cause does not perish, being permanent, 
but continues to exist and operate, it continues to 
produce more and more of the effect, and thus we 
get a progressive effect, from the accumulating influ- 
ence of a single permanent cause. 

III. This peculiar case is evidently only a 
case of the composition of causes, — the cause 
being here compounded with itself. 

IV. There are two cases or kinds of Progres- 
sive Effects : — 



152 ON PROGRESSIVE EFFECTS. 

1. When the cause though constantly acting is 

not variable. 

2. When the constantly-acting cause itself varies. 

In the second case, where the cause itself is variable, it 
is clear that it may be regularly or irregularly so, 
and if the former it may be simply progressive or 
pass through a cycle of changes. In such cases the 
effect is progressive, as in the former case, but not 
regularly progressive, the quantities added to the 
effect in equal times are not equal. 

V. How Progressive Effects are logically in- 
vestigated. 

We have already seen that cases of composition of causes 
can seldom satisfactorily be investigated, except by 
the Deductive Method ; and this is pre-eminently 
true in the case of Progressive Effects, since the 
continuance of the cause influences the effect only 
by adding to its quantity ; and since this addition 
takes place in accordance with a fixed law, the re- 
sult can be computed mathematically, — the most 
complete example of the Deductive Method. 

VI. Most uniformities of succession, which 
are not cases of causation (i.e., a series of two or 
more terms in which each term is not caused by 
its predecessor) are cases of Progressive Effects. 

In all cases of Progressive Effects there will evidently be 
a uniformity of succession between any stage of the 
effect and the next succeeding ; thus, the second 



EMPIRICAL LAWS. 153 

particle of rust on iron succeeds the first, the third 
the second, and so on. And, generally, whenever 
we find any phenomena going through a regular 
process of variation, we do not presume that any 
term of the series is the effect of its predecessor, 
but rather that the entire series originates from the 
continued action of some permanent cause, — that, 
in a word, it is a Progressive Effect. 



CHAPTEK XVI. 
Empirical Laws. 

I. An Empirical Law is an observed uniformity, 
presumed to be resolvable into simpler laws, 
but not yet resolved into them ; or it is a law 
whose why has not been ascertained. 

The distinction between Ultimate and Derivative Laws 
has been already explained. Empirical Laws belong 
to the class of Derivative Laws, and constitute that 
section of them which has not been resolved into 
any simpler laws ; thus — 



Resolved. 



Derivative ! 



f Known to be cases of 



Y ¥.fr, c .i ro ^ J causation. 



Laws. I Not resolved 



I Not known to be cases 
L of causation. 



154 EMPIRICAL LAWS. 

We have thus two kinds or classes of Empirical Laws, 
and although all unresolved Derivative Laws may 
be termed Empirical, yet the designation, in its 
strictest sense, belongs to those not known to be cases 
of causation. For example, — let the Derivative Law 
assert that A is followed by a ; if it belong to first 
class of Empirical Laws, we should know that A is 
the cause of a (but not why it is the cause) ; in 
the second class we should not know even that 
much. 

II. Derivative Laws mostly depend upon 
collocations. 

Since, when a Derivative Law is resolved it is usually 
found to be derived or to result from two or more 
simpler laws, it is evident that the fulfilment of the 
Derivative Law not only depends upon the existence 
of the simpler laws, but also upon their existence 
together, so as to intermix their effects. That is, 
the simpler agencies must be collocated, or arranged, 
in a particular manner. 

III. Collocations of causes cannot be re- 
duced to any law. 

We cannot, in general, lay down any fixed principles 
applicable to the mode of arrangement or distribu- 
tion of natural agencies. 

IV. From this it arises that Empirical Laws 
cannot be relied on beyond the limits of actual 
experience. 



EMPIRICAL LAWS. 155 

This property is highly characteristic of Empirical 
Laws. 

A Derivative Law which is resolvable wholly into a 
single more general law will be as universally true as 
that one law ; but when it depends upon several 
such laws, we have seen that those laws must coexist 
in a certain manner, or the Derivative Law will not 
be fulfilled. Now it is the very nature of a Deriva- 
tive Law which has not been resolved, that we do 
not know whether it results from one law, or more 
than one ; and if the latter, what collocations are 
necessary. Hence, we cannot be sure that the un- 
resolved law will be found true, beyond the limits 
within which it has actually been found true, and 
where the necessary conditions are known to prevail. 

V. Generalisations which rest solely on the 
Method of Agreement can only be received as 
Empirical Laws. 

As already explained, we can never prove causation by 
this method ; all that it proves is, that two pheno- 
mena {A and a) are found together, — an Empirical 
Law; we can never be sure that some unknown 
antecedent (B) is not either the cause of both 
phenomena, or causes one (a) while having been in 
our experience invariably conjoined with the other 
{A). 

Thus, suppose we have two phenomena, — redness of sky 
in the morning (A), and a fall of rain (a), these 
phenomena may be related in the following ways : — 

A — a A being real cause of a. 

g ^A ( A and a both produced by some unknown 

a \ cause, B. 
A f Some unknown cause, B, produces a, and 

B — a\ has been always conjoined with A, 



156 EMPIRICAL LAWS. 

We may, however, add that in proportion as we have 
reason to suspect that the Empirical Law does not 
depend upon collocations, so may we rely more con- 
fidently upon it, in extending it to new cases. And 
also the more general an Empirical Law is, so is it 
the more certain. 

VI. Signs by which an unresolved law may 
be presumed to be derivative and not ulti- 
mate. 

1. Indications of any intermediate link between 

the antecedent and consequent. 

If the effect is of such a kind that we see that it 
is probable that the obvious cause is not the 
immediate cause, we may infer that the law 
which connects such an effect with its obvious 
cause is resolvable. 

2. Complexity of the antecedent. 

In such a case we may infer that more than one 
of the elements which constitute the antecedent 
are concerned in producing the effect ; and that 
therefore such effect is produced by a conjunc- 
tion of causes, and the law of its production is 
therefore resolvable into the simpler laws of 
the causes which concur in generating it. 



ELIMINATION OF CHANCE. 157 



CHAPTER XVII 
Elimination of Chance. 

I. Proof- of Empirical Laws depends partly on 
Theory of Chance. 

The method of agreement per se, can only show that two 
phenomena are conjoined; and owing to plurality of 
causes, it does not prove that these conjunctions are 
anything more than mere unconnected coincidences, 
until the process for the elimination of chance has 
been applied. 

As an example of a circumstance which is always coin- 
cident with phenomena on the earth's surface, and 
yet has nothing to do with many of these, — take 
gravity, or the magnetic influence of the earth. 

II. Definition of Chance. 

Chance only applies to conjunctions (se- 
quences or coexistences) of phenomena; and 
conjunctions of phenomena are said to happen 
casually or by chance, when those phenomena 
are in no way related through causation. 

No phenomenon, or event, can properly be said to be 
produced by chance, — i.e., immediately produced. 



158 ELIMINATION OF CHANCE. 

Given the cause or combination of antecedents, and 
the effect necessarily follows. But when an event 
is said to be produced by chance, what is really 
meant is, that the conjunction of antecedents or 
conditions upon which that phenomenon followed, 
happened without any causal connexion between 
them. 
Take this case, — the particular position of Jones in a 
battle, and the particular path of a bullet in that 
battle, killing Jones by shooting him through the 
heart. Now Jones' death was an effect necessarily 
consequent upon his heart being in the path of the 
bullet : so far the effect followed its cause without 
any question of chance : but the conjunction of 
Jones' particular position and the particular path 
of the ball might be wholly casual. 

The Elimination of Chance is applied in two dis- 
tinct cases: — 

1. In the case of conjunctions of phenomena (as just 

explained). 

2. In the case of a constant cause associated with casual 

causes ; and this again presents two cases, as we 
shall see. 

III. Application of this process to conjunc- 
Dns of phenomena. 



tions of phenomena. 



The question we have to solve is, — After how many and 
what sort of instances may it be concluded that 
an observed coincidence (conjunction) between two 
phenomena is not the effect of chance ? 

This question is answered through another, — Do the 
coincidences occur more (or less) frequently than 



ELIMINATION OF CHANCE. 159 

chance will account for 1 Are the two phenomena 
more or less frequently conjoined than would occur 
on the supposition that no connexion exists between 
them? 

No general answer can be given to this inquiry ; 
all we can do here is to point out the principle upon 
which ive must proceed in eliminating Chance in 
cases of conjunctions of phenomena : — 

We consider first the absolute frequency of each 
phenomenon itself ; and then how great frequency 
of conjunction must therefore be expected without 
supposing any connexion (either of tendency to 
cause or of tendency to prevent the one or the 
other) between them. If we actually find them 
conjoined more frequently than this, we may (with 
certain precautions which we will pass over for the 
present) infer that there is some connexion ; if less, 
often that there is some repugnance. 

Thus, suppose we are considering whether there is any 
connexion between a red sky and rain, we should 
first determine the absolute frequency of each 
phenomenon. Suppose we find that a red sky 
occurs one day in three, and rain one day in two, 
in every six days there would be a casual con- 
junction between them, and half the number of red 
skies would be in that way associated with rain. 
This, then, is the frequency of conjunction due to 
chance, with which we have to compare the observed 
frequency of conjunction. 

IY. Application of the process in the case 



160 ELIMINATION OF CHANCE. 

of a constant cause associated with casual 
causes. 

As an example of this case, — the position of the sun 
is the constant (i.e., for a few days) cause of the 
temperature of any given day, but with the effect 
of this are blended the effects of many casual 
causes — wind, clouds, &c. 

There are two cases of this combination : — 

1. Where the effect of the constant cause forms so 

great and conspicuous a part of the total result 
that its existence as a cause could never be a 
matter of uncertainty. 
Here the elimination of chance determines how much 
of the total effect is due to the constant and casual 
causes respectively. 

2. Where the effect of the constant cause is so small 

compared with that of the changeable or casual 
causes, that its very existence as a cause may be 
unknown. 
Here the process actually determines the existence of 
the constant cause as a cause. 

Case 1. — Where effect of constant cause is large 
compared ivith total effects of casual causes. 

Suppose a chemist to weigh repeatedly the same body, 
the actual result which he obtains in any one ex- 
periment is mainly determined by the true weight of 
the body (this is the conspicuous constant cause A)\ 
but, besides this, the result would be modified in 
each ease by casual causes, — sources of error, — 






ELIMINATION OF CHANCE. 161 

draughts, imperfections of the balance, &c, which 
would vary in each experiment. Now if 

(a.) We could completely isolate the constant cause A 
(i.e., in this case remove all sources of error), we 
could, of course, directly determine the part of any 
given effect due to it ; but if not, 

(b.) We must apply the process of eliminating the effects 
of the casual causes in the following manner : — 

We make as many trials as possible, A being 
preserved invariable ; the results of these different 
trials will naturally differ, but (if, as we here assume, 
the casual causes in the long run tend as much for 
as against the constant cause) in such a manner as 
to oscillate about a certain point, sometimes being 
greater, sometimes less. If so, we may conclude 
that that mean or average result is the part due to 
our constant cause A ; the variable remainder being 
the effect of the casual causes. 

The test of the sufficiency of this Induction is, that 
any increase in the number of trials does not 
materially alter the average. 

Case 2. Where the effect of the constant cause is 
comparatively inconspicuous, — in such a case we may 
discover a residual phenomena {i.e., constant cause) 
by the elimination of chance. 

This is merely a particular case of the process just de- 
scribed, but here the very existence of the constant 
cause is determined by the same method which, in 
the previous case, served to ascertain the quantity 
of its effect. 
This case of Induction may be characterised thus : — A 
given effect is known to be chiefly, and not known 
not to be wholly, determined by casual causes. If, 
indeed, it be wholly so produced, then the average of 

L 



i 



162 ELIMINATION OF CHANCE. 

the effect will be zero, the effects of the casual 
causes cancelling each other. If, however, the 
average be not zero, but some positive quantity, 
about which the total effect oscillates equally, — 
sometimes above, sometimes below, — we may con- 
clude that this quantity is the effect of some 
constant cause, which cause we may set about 
detecting. 

Thus, a slightly loaded die may be detected in 
this way ; the loading is a slight permanent cause, 
mixed with casual causes. If the die were fair, in 
a large number of trials, we should get about the 
same number for each face, but if there is a steady 
preponderance in any particular number, we may be 
sure that the die is proportionally loaded. 

V. We know, if we make a very large number of throws 
with a fair die, that each of the six faces of the 
die will come uppermost about the same number 
of times ; but if we make only a limited number of 
trials, there may be considerable deviation from this 
average; for instance, in four trials, the six may come 
up twice or even three times. In like manner, in 
reference to the coincidences of phenomena which 
we are now discussing, besides the question, 

"What is the number of coincidences which on an average 
of a great number of trials (or in the long run) may 
be expected to arise from chance alone ? which has 
been considered in this chapter, there is another : — 

Of what extent of deviation from that average is the 
occurrence credible from chance alone in some 
limited number of instances? This question is 
answered by the doctrine of Calculation of Chances, 
which is next considered. (See Chap. XVIII., Sub- 
division V.) 



CALCULATION OF CHANCES. 163 



CHAPTEK XVIII. 

Calculation of Chances ; or, Theory of 
Probability. 

I. The probability of a given event to any 
one is the degree of expectation of its occur- 
rence which he is warranted in entertaining 
by present evidence. 

It is very necessary to remember that the probability of 
an event is not a quality of the event itself, but 
merely a name for the degree of ground which some 
individual has for expecting its occurrence. Every 
event is in itself certain to occur or not to occur, 
and if we were omniscient, there would be no such 
thing as probability. 

There are two forms of Probability : — 

1. Simple probability without specific knowledge (i.e., of 

the comparative frequency with which the different 
possible events in fact occur). 

2. Probability based more or less upon such specific 

knowledge. 

II. The first form of Probability is charac- 
terised thus, — " out of several events we must 
know that some one will certainly happen, and 



164 CALCULATION OF CHANCES. 

one only ; and we must not know, nor have 
any reason to expect that it will be one of 
these rather than another." 

Suppose we are required to take one ball from a box, of 
which we only know that it contains black, white, 
and red balls, and none of any other colour. Here 
we know that either a white, or black, or red ball 
will be drawn, but we have no reason for expecting 
one rather than the other. In such a case, the 
drawing of any particular colour is equally probable 
to us, and it will be indifferent upon which we stake ; 
while, if we stake against any colour, the odds are 
two to one against us. This is an example of the 
first kind of probability ; but if we have* some spe- 
cific knowledge of the actual frequency with which 
white, black, and red balls were drawn, if, for in- 
stance, several trials had previously been made, or 
if we knew that there were more black than white 
or red balls, it would become a case of the second 
kind, — that is, we should have some reason to expect 
one event rather than the other. 

The basis of this first kind of probability is clearly the 
general and axiomatic principle, that out of the pos- 
sible cases there must be a majority against each, 
except one at most ; and since we cannot presume 
any one to be in this position, we have no ground 
for electing one rather than another. This principle 
being universal, there is, in our reasoning, no refer- 
ence to specific knowledge, — knowledge special to the 
particular case. 
Although the first kind of Probability is interesting as a 
mathematical study, yet the practical probability 
with which Logic and non-mathumatical sciences are 



CALCULATION OF CHANCES. 165 

concerned belongs almost exclusively to the second 
kind, and this for two reasons, — (1.) because, as re- 
gards the first kind, in practical questions it is 
usually impossible to make out the list of events 
which are possible ; and (2) because it is very rare 
that we cannot obtain some specific knowledge which 
will immensely aid us in our calculation. 

III. In the second kind, therefore, with 
which we are here concerned, our conclusions 
respecting the probability of a particular fact 
rest upon more or less knowledge of the pro- 
portion between the cases in which it occurs 
and in which it does not occur; this know- 
ledge being either derived from specific ex- 
periment, or by reasoning upon the causes in 
operation which tend to produce, compared 
with those which tend to prevent, the fact in 
question. 

The basis then of our estimate of this kind of probability 
is either an Induction or a Deduction, and it is very 
necessary to notice that it is not unimportant 
which. If on trial we find that we draw one black 
ball on an average to every nine white ones, the con- 
clusion that an event occurs once in ten times is as 
much an Induction as that the event occurs uni- 
formly. But when we arrive at a conclusion in this 
way by simply counting instances in actual experi- 
ence, we only get an empirical law, we know 
nothing of the " why." But if we know the causes 
which operate for or against, we can deduce the pro- 



166 CALCULATION OF CHANCES. 

bability in a much more certain manner. Indeed, 
it is really by a Deduction that we infer that from 
an urn containing ten black and ninety white balls, 
we may nine times as much expect to draw a white 
ball as a black, because we consider that the hand 
may alight in nine places and find a white, and in 
only one and find a black. So a betting man does 
not rest content with the results of actual trials of 
the horses, but examines the animals separately, to 
get as much as possible at the causes of superior 
speed. This, too, is the reason why the first occur- 
rence of an event, about the possibility of which 
there might be a doubt, adds so much more to its 
probability than any subsequent happening of it 
does, because we are thereby assured that causes 
really exist adequate to produce it, which till it 
actually occurred we might not be certain of. 

Still, however, as a matter of fact, in most cases 
in which the estimation of chances is applied to 
practical uses on a large scale, the data are drawn 
not from knowledge of the causes, but from direct 
experience of the relative frequency of events of the 
different kinds. The chances of life, of recovery 
from disease, of accident, shipwreck, &c, are drawn 
from registers of the actual occurrence of those 
phenomena, — i.e., from the observed frequency not 
of the causes but of the efects themselves. 

IV. Theory of probability which relates to 
the cause of a given event (effect). 

The question is — Which of several causes is 
most likely to have produced a given effect \ 

Answer. — Given an effect to be accounted for, and there 



CALCULATION OF CHANCES. 167 

being several causes which might have produced it, 
but of the presence of which in the particular case 
nothing is known ; the probability that the event 
was produced by a given one of these causes is as 
the antecedent probability of the existence of that 
cause, multiplied by the probability that that cause, 
if it existed, would have produced that effect. 

V. Given the average number of coinci- 
dences to be looked for between two uncon- 
nected phenomena, — What are the chances 
of any given deviation from that average ? 
(See end of last chapter.) 

We observe a certain number or succession of coinci- 
dences between A and B ; if J. and B are merely 
coincident by chance, we know how many such coin- 
cidences ought to be expected ; the question then 
follows what are the chances that the observed coin- 
cidences between A and B are merely casual ? If 
not casual, they must be the result of some law. 

To answer this we compare the two probabilities : — (1.) 
the probability that the coincidences are due to 
chance, and (2.) the probability they are due to 
some law. 

As to (1.), if the known probability of a single coinci- 
dence be — the probability that the same coinci- 
dence will occur n times in succession is — * 

m n 

As to (2.), the probability that the coincidences are due 
to some law, admits of more or less exact estima- 
tion according to circumstances : — 

(a.) We may know what the cause of the conjr.uc- 



168 EXTENSION OF DERIVATIVE LAWS. 

tion must be if a cause exist at all, and we 
may be able to estimate the probability of its 
presence. 
(6.) But if we do not know any known cause which 
would account for it, it is clear we cannot 
estimate the probability of the presence of 
an entirely unknown thing, and we are driven 
to the result of the first question, — are the 
conjunctions so successive or numerous that 
their being produced by chance would be a 
very uncommon thing ? If so, since it is not 
a very uncommon thing to discover a new 
cause in nature, we may conclude that such 
does here operate in causing the conjunc- 
tions, and may lay down provisionally an 
empirical law. 



CHAPTER XIX. 

Extension of Derivative Laws to Adjacent 

Cases. 

I. Derivative Laws are less general and less 
certain than the Ultimate Laws into which 
they are resolvable, or presumed to be resolv- 
able, for two classes of reasons : — 

(1.) They very often depend upon collocations (co-exist- 
ence in a particular manner) of causes for which 
there is no law. 

(2.) They are more apt to be counteracted ; and for one 






EXTENSION OF DERIVATIVE LAWS. 169 

phenomenon to be produced independently of the 
other. 

This has already been explained of the different forms 
of derivative laws (see page 140), and we only 
remark again here, that every derivative law which 
is resolvable into or derived from two or more 
simpler laws, will only be fulfilled where the two 
simpler laws are found together, — i.e., its fulfil- 
ment depends upon collocations. Thus the rise of 
mercury in the Torricellian tube is a derivative 
law, resolvable into two simpler laws, gravity and 
equal transmission of liquid pressure. Now if 
these two laws do not act together (as when the 
Torricellian experiment is made in a vacuum), the 
derivative law (that the mercury will rise) is no 
longer fulfilled. So also the immense majority of 
derivative laws depend partly upon collocations. 

Again, when we have two effects conjoined, arising from 
the same cause, say a and b from the cause c, it 
may happen that either a or b may be produced 
alone by some other cause than c, and then in that 
case a and b will not be found together, and the 
law which asserts that a and b are always conjoined 
will fail. 

II. From all this it follows that Derivative 
Laws (and especially unresolved Derivative 
Laws) must only be extended to cases adjacent 
(i.e., contiguous or similar) in Time, Place, and 
Circumstances. 

1. As regards Time, take the derivative law that day and 

night succeed each other. 
Now we extend this law with confidence to cases 



170 EXTENSION OF DERIVATIVE LAWS. 

adjacent in time, — we expect that day and night 
will succeed each other for some time to come. In 
this case we know the causes, — the derivative law 
has been resolved, — they are the opaque earth 
rotating upon an axis, lying in a certain direction, 
and the sun shining. Now, as long as these causes 
exist, and are not counteracted, the derivative law 
will not fail ; but we know by observation that 
these phenomena have continued unaltered for at 
least five thousand years, and, therefore, during 
that time no counteracting cause has diminished 
them in any appreciable degree. And it is opposed 
to all experience that such a cause should start into 
immensity in a single day, or in a short time. 

If we did not know the causes of day and night, we 
could still draw a similar conclusion with less ex- 
tension as to the future. We should know at least 
that the phenomena had been for five thousand 
years conjoined, and we might therefore infer that 
the causes had not been counteracted during that 
period, when the same conclusions would follow as 
in preceding case. But still we could not be 
assured that if we did know the causes we could 
not predict their destruction from agencies actually 
in existence. Thus, a clock has been made to go 
for years without interference ; a savage wholly 
ignorant of its mechanism, seeing it going on 
steadily day after day might easily imagine its 
movement perpetual, but a person who knows the 
causes, knows that it contains within itself the 
causes of its own cessation. 

In either case the argument becomes weaker in propor- 
tion as wc extend the period which our prediction 
covers. 

2. As regards Place, it might seem that an empirical 



ON ANALOGY. 171 

law could not even be extended to adjacent cases, 
for the existence of a cause in any one place is no 
guarantee of its existence in any other place. 
When, therefore, we extend such a law to cases 
beyond the limits of place in which it has been 
observed, such cases must be presumably within 
the influence of the same individual agents. 



CHAPTER XX. 

On Analogy. 

QIt is very necessary to clearly understand and 
keep in view the different senses in which the term 
" Analogy " may be used in different cases. 

The general formula of inference from experience 
may be given thus : — 

A certain object (or set of objects), A, having a certain 
property (or set of properties) x, have also a certain 
property m; 

Another object, B, resembles A in possessing x ; 

•■. B also resembles A in possessing m. 

If for A we substitute ''certain animals," for x "split 
hoofs," for m " rumination," and for B " some par- 
ticular animal," the form of the argument will be 
seen. 



172 ON ANALOGY. 

Now, this general formula includes several par- 
ticular forms of reasoning : — 

1. Analogy in the strict sense. 

2. Example. 

3. Imperfect Induction. 

4. Perfect Induction. 

1. Analogy in the strict sense, or resemblance of rela- 

tions. 

2. Example. — The characteristic of this is, that between 

x (the properties which constitute the resemblance 
between A and B), and m (the property inferred to 
exist in B), no connexion whatever is known, nor 
are they known to be not connected. 

Thus, let A be " the earth ; " x spherical shape, 
rotation on axis and atmosphere ; m being inhabited, 
and B Venus, then the argument of Example stands 
thus : — 

The earth possesses spherical shape, &c, and is 
also inhabited. 

Venus resembles the earth in possessing the for- 
mer set of properties. 

•■■ Venus is inhabited. 

Now, in this case, no connexion whatever is known to 
exist between the properties which form the resem- 
blance and the inferred property, " being inhabited," 
and we do not know that they arc unconnected with 
it. This form, therefore, is the argument from 
simple or mere resemblance. 

3. In Imperfect Inductions some connexion is shown 

between the resembling properties and the inferred 



ON ANALOGY. 173 

property, short of the former being a mark of the 
latter. 
4. In Perfect Inductions we show, by a due comparison 
of instances, that x is a mark of m, that the proper- 
ties which constitute the resemblance are invariably 
accompanied by the inferred property. This con- 
stitutes the distinction between "Analogy" and Per- 
fect Induction. 

Mill does not make use of the term " Example" but, 
after discussing " Analogy Proper" he goes on to 
consider what is here called " Example" under the 
general head of Analogy ; and this plan we shall 
follow here to avoid deviating from his phrase- 
ology. 2 

I. Analogy in the strict sense. 

Here the two resembling things (our A and B) are rela- 
tions, as we might say, the relation between a despot 
and his people resembles the relation between a 
father and his family. 

The condition of the validity of this form of argument is 
this, — that the two relations should really resemble 
each other in that particular fact or circumstance 
upon which the inferred property depends. 

II. Analogy in general. 

The argument from Analogy amounts to this, — a pro- 
perty, m, known to belong to A, is more likely to 
belong to B, if B agrees with A in some of its pro- 
perties (though we know no connexion between m 
and any of these properties), than if no resemblance 
whatever could be traced between B and A. 



174 ON ANALOGY. 

It is requisite to an analogical argument of this hind — first, 
that the property, m (being inhabited), shall not be 
known to be connected with any of the common 
properties of A and B (the earth and Venus) ; and, 
secondly, m must not be known to be unconnected 
with any of these common properties, or such clearly 
count for nothing in i^ie argument. 

Every additional resemblance (in points not known to be 
unconnected with m) between A and B, so far 
favours the conclusion that B possesses m like A. 

Every additional dissimilarity, in like manner, between 
A and B weakens by so much the conclusion. 

The value of an analogical argument, then, de- 
pends upon three things : — 

l*. The amount of ascertained resemblance between 
A and B. 

2. The amount of ascertained dissimilarity between 

them. 

3. The amount of unascertained properties in A 

amdB. 

Such an argument may, therefore, come very near 
to a valid Induction, if — 

1. The resemblance is very great. 

2. The dissimilarity very small. 

3. Our knowledge of the subject-matter tolerably 

extensive. 

Suggestive use of Analogy — 

Is this respect analogical considerations have often the 
greatest scientific value ; there is no Analogy, how- 
ever faint, which may not be of great importance in 



EVIDENCE OF THE LAW OF CAUSATION". 175 

suggesting experiments or observations, which may 
lead to more positive conclusions. 

£±11. Imperfect Inductions. 

Here we show that some connexion, short of actual 
causation (which would be a Perfect Induction) 
exists between the properties which constitute the 
resemblance and the inferred property, m. We 
may show that the former is an important part of 
the cause of the latter, or that it has a tendency to 
prevent the existence or effects of counteracting 
causes, &c. &c.J 



CHAPTEK XXI 

Evidence of the Law of Causation. 

I. The Inductive Methods presuppose the 
universality of the Law of Causation ; if we 
believe that any phenomenon can start into 
being without a cause, the conclusion from 
any one of them fails at once. 

Since, then, this great law lies at the bottom of our In- 
ductive inferences, upon what evidence is it itself 
based 1 



176 EVIDENCE OF THE LAW OF CAUSATION. 

II. Answer of the Intuitive School. 

They say that the truth of the law is certain because 
the mind cannot help believing it ; it is an intuitive 
truth, acquiescence in which is necessitated by the 
laws of the thinking faculty. 

Mill replies thus : — 

In opposition to this view I must reiterate my protest 
against adducing as evidence of the truth of a fact 
of external nature any necessity the human mind 
may be supposed to be under of believing such fact. 
It has already been shown (page 77, &c.) that such 
conceivability or inconceivability of anything de- 
pends mainly upon the mental history of the person 
making the attempt. Moreover, in this case, the 
statement is not true ; there is no difficulty in con- 
ceiving that phenomena may start into being with- 
out a cause, for many philosophers believe that the 
impulses of the will are of this character, and the 
ancients recognised (l chance" and "spontaneity" 
amongst natural agencies. 

III. The Law of Causation is really proved 
by an Induction per enumerationem sim- 

plicem. 

As men recognised one instance of causation after 
another, they would first suspect that many effects 
have a cause ; as experience widened, that most 
events have; and finally, as an immense number of 
experiences accumulated, without any certain ex- 
ception, they would believe that the law was uni- 



EVIDENCE OF THE LAW OF CAUSATION. 177 

versa!, perhaps really before, in logical strictness, 
they were warranted in so doing. 

IV. Under what conditions Induction per 
enumerationem simplicem, is a valid process. 

We have already laid down that this method is valid, 
precisely in proportion to our assurance that if an 
exception ever did occur we should know of it. In 
other words, 

Precisely as the subject-matter is limited and special, so is 
the process insufficient and delusive. 

As its sphere widens, this unscientific method becomes 
less and less liable to mislead ; and the widest 
truths, viz., the law of Causation, and the primary 
laws of number and extension, are proved by this 
method alone, nor are they susceptible of any other 
proof. 

Induction by simple enumeration leads to Empirical 
laws ; these cannot be extended beyond adjacent 
cases, because the causes may cease to exist or be 
counteracted, or the requisite collocations may fail. 
If, however, we suppose the subject-matter of our 
law to be so widely diffused that there is no time, 
place, or set of circumstances in which it is not ful- 
filled, it is clear that the law cannot be frustrated 
by any counteracting causes except such as never 
occur, and cannot depend upon any collocations ex- 
cept such as exist at all times and places. It is 
therefore an Empirical law co-extensive with human 
experience, at which point the distinction between 
Empirical laws and laws of nature vanishes, and the 
proposition takes its place amongst the highest 
order of truths accessible to science. Such is the 
Law of Causation. 

M 



178 EVIDENCE OF THE LAW OF CAUSATION. 

V. The Law of Cause and Effect beino; thus 
certain, is able to confer the same certainty 
upon all laws which can be deduced from it. 

The utmost we can do in the way of proof for any law is 
to show that it is true, or the Law of Causation is 
false, — that a phenomenon has the assigned cause 
or none. The Inductive Methods are formulae of 
the modes of doing this. 

VI. Summary of proofs which we now have 
of the universal and absolute truth of the Law 
of Causation. 

1. We know it to be true of by far the greater number 

of phenomena. 

2. There are none of which we know it not to be true . 

3. Of those phenomena of which we do not positively 

know it to be true. 

(a.) One after another is constantly passing 
from this class into that of known examples 
of its truth, as they are better understood. 
(b.) And the deficiency of our positive knowledge 
with respect to these phenomena may always 
be accounted for by their rarity or obscurity. 
(<-.) And finally, every such phenomenon, pro- 
duced apparently without cause, obeys in 
some other respects known laws of nature, 
and therefore we may presume this law also. 
We may, however, add that these reasons do not hold 
for the prevalence of this law beyond the limits of! 
our experience, in distant stellar worlds for example. 



COEXISTENCIES INDEPENDENT OF CAUSATION. 179 

CHAPTEK XXII. 

Coexistences independent of Causation. 

I. Properties of objects may be divided into 
two classes : — 

1. Those which are results of Causation (derivative). 

2. Those which are ultimate, — which are not results 

of Causation, which are not connected with ante- 
cedent phenomena by any law. 

Thus, if we contemplate the substance " oxygen," we 
find it to present numerous properties ; some of 
them are manifestly derivative, depending upon 
assignable causes ; thus, its gaseous form is pro- 
bably due to latent heat ; but after subtracting all 
such, there remains a number which seem inherent 
in oxygen, which are themselves the causes of other 
properties, but are not themselves caused by any ; 
these are ultimate properties. 

It must not be supposed that we are always able to 
determine whether a given property is really ulti- 
mate or not ; very often we know positively that it 
is derivative, and in no case can we be certain that 
it is not. Still, for practical purposes, we regard 
those properties as ultimate, which we have no 
reason to suppose derivative. 

II. The coexistences, then, with which we 
are now concerned, are the coexistencies of 



180 COEXISTENCIES INDEPENDENT OF CAUSATION. 

these ultimate properties ; A and B are found 
together, but their coexistence cannot be ac- 
counted for by Laws of Causation, — we know 
no reason why they should, but only that they 
do coexist. 

III. Mode in which propositions assertive 
of coexistence of ultimate properties are 
proved. 

Propositions expressive of such coexistences are to be 
regarded as Empirical laws, and the only proof of 
which they are susceptible is by the Induction by 
simple enumeration. 

The reason why we are thrown back upon this method 
is, because there is no general axiom bearing the 
same relation to uniformities of coexistence as the 
law of Causation bears to uniformities of succession. 
If B follows A, and we can show that A is the cause 
of B, then we may be sure that where A is present 
(counteracting causes being absent) B will also be 
found, — A will be a mark of B. But if A and B are 
found to coexist, and their coexistence cannot be 
traced to causation, we have no similar axiom to 
give us an assurance that they will be invariably 
coexistent, — that where A is, B also will always be 
found. The overlooking of this distinction was the 
grand error in Bacon's system of philosophy. 

IV. Empirical Laws are stronger in propor- 
tion as they are more general. 

The condition of the validity of the method of Induction 
by simple enumeration has already been several 



APPROXIMATE GENERALISATIONS. 181 

tiroes pointed out ; as its sphere widens it becomes 
more and more trustworthy, and in the last chapter 
it has been shown that there is a point of generality 
at which Empirical laws become as certain as laws 
of Nature, — or rather there is no longer any distinc- 
tion between them. 

For (1.) if an Empirical law be really a law of Causation, 
the more general it is, the greater is proved to be 
the space over which the necessary collocations pre- 
vail, and within which counteracting causes do not 
exist ; and (2.) even if not a result of Causation, but 
expressive of an ultimate coexistence (as we here are 
specially considering), the more general the law is, 
the greater amount of experience it is derived from, 
and the greater the probability therefore that if excep- 
tions ever occurred, we should know of them. 

Hence it requires stronger evidence to establish an ex- 
ception to a law of this kind, in. proportion to its 
generality. 



CHAPTER XXIII. 

Approximate Generalisations and Probable 
Evidence. 

(An Approximate Generalisation is one of the form " Most 
A is Br) 

I. When a conclusion is said to rest upon 
mrobdble evidence, the premisses from which it 



i 



182 APPROXIMATE GENERALISATIONS. 

is drawn are usually approximate generalisa- 
tions. 

As every certain inference implies that there is ground 
for a proposition of the form " All A is B" so every 
similar probable inference implies that there is 
ground for an assertion of the form " Most A is B ; " 
and the degree of probability in such a case will 
depend upon the proportion between the number of 
instances which agree with, and the number which 
conflict with, the generalisation. 

II. Approximate generalisations are. of much 
less value for the purposes of Science than for 
the purposes of practical life. 

Why of inferior value in Science ? 

Because, beside the inferior precision of such proposi- 
tions, and beside the inferior assurance with which 
they can be applied to particular cases, they are 
almost useless as a means of discovering ulterior 
truths by way of Deduction. In a Syllogism which 
contains an approximate generalisation as a premiss, 
we can only at the very utmost obtain another 
approximate generalisation as a conclusion, and 
generally no conclusion at all. 

Their use in Science, then, is chiefly as sugges- 
tions of, and materials for, universal truths. 

Why of greater value as practical guides? 

Because (1.) they are often our only resource ; (2.) the 
laws of phenomena, even when known, are commonly 



APPROXIMATE GENERALISATIONS. 183 

too much encumbered with conditions to be adapted 
for every-day use ; and (3.) our decision is often re- 
quired so rapidly that we are compelled to act upon 
probability, without waiting for certainty. 
The principles of Induction, therefore, applicable to ap- 
proximate generalisations are not a less important 
subject of inquiry than the rules for arriving at 
universal truths, but they are mere corollaries from 
these latter. 

III. There are two classes of cases where 
we are forced to be guided by approximate 
generalisations : — 

1. When we have nothing better ; when we have not 

been able to carry our investigations of the laws of 
the phenomena any further. 

As an example we may give Newton's generalisa- 
tion — Most highly refractive substances are 
combustible. 
The importance of this class is not very great. 

2. When the generalisation (Most A is B) is not the 

ultimatum of our scientific knowledge, but the 
knowledge we really possess beyond it cannot be 
conveniently brought to bear upon the particular 
case. 

In such cases we really know what circumstances distin- 
guish those A's which possess B from those which 
do not, but we have not the time or means to ex- 
amine whether such distinguishing circumstances 
exist in the present case or not. 

This is usually the case with inquiries of the kind called 
" moral ;" i.e., having for their object the prediction 
of human actions. Nearly all propositions relating 



184 APPROXIMATE GENERALISATIONS. 

to these must be thrown into the form—" Most 
persons act so and so in such and such, circum- 
stances." Not but that in general we know well 
enough upon what internal dispositions and external 
circumstances the actual result will depend, but we 
have seldom the means of knowing whether and how 
far any given individual possesses those internal 
qualities, or is under the influence of those external 
circumstances. 

We may, therefore, divide Approximate General- 
isations into two Classes : — 

1. Where they embody the total or ultimatum of our 

knowledge. 

2. Where they do not (but constitute the most available 

form for practical guidance). 

IV. How approximate generalisations are 
proved: — 

It is necessary to consider separately the two forms 
of these laws : — 

1. Where the approximate generalisation in question 
comprises all we know of the subject. 

We know only that " Most A is B ;" not why they 
are so ; nor in what respects those which are differ 
from those which are not. 

In this case we arrive at " Most A is J5," in pre- 
cisely the same manner as at "All A is B," if the 
latter happened to be the truth in the case. We 
collect a number of instances sufficient to eliminate 
chance conjunctions of the phenomena A and B. 



APPROXIMATE GENERALISATIONS. 185 

and having done so we compare the number of cases 
where A possesses B with the number of cases 
where it does not, and frame our conclusion accord- 
ingly- 
2. Where the approximate generalisation is not the ulti- 
matum of our knowledge. 

When we know not only that "Most A is B" but 
also the causes of B, or some properties by which 
the portion of A possessing B is distinguishable 
from the portion of A without B (criteria of 
B). 

In this case we have a double mode of proving 
that " Most A is B ;" (1.) the Direct, as in the pre- 
ceding case ; and (2.) the Deductive or Indirect, — 
examining whether the proposition can be deduced 
from the known causes or known criteria of B. 

[Example : — Suppose the proposition to be proved 
is " Most Scotchmen can read ;" by the direct mode 
we should examine a sufficient number of Scotch- 
men, and deduce our conclusion accordingly ; by the 
indirect, knowing that a cause of " being able to 
read" is " being sent to school," and a criterion of it 
would be the actual reading of a newspaper, we 
might inquire what proportion of Scotchmen were 
sent to school, or to what extent newspapers, &c, 
were circulated amongst them.] 

It is evident that in cases of this class we may 
always substitute a universal proposition for the 
approximate, by introducing the cause or criterion 
as a qualification. Thus — " All Scotchmen, who 
have been properly taught, can read." 

V. Precautions in arguing from approximate 
generalisations to individual cases. 



186 APPROXIMATE GENERALISATIONS. 

1. As to the direct application of an approximate gene- 

ralisation to a single instance. 

Approximate generalisations, being Empirical 
laws, can only be relied upon as regards cases 
presumably within the limits of time, place, and 
circumstances of our observation. 

If the proposition "Most A is B" has been suf- 
ficiently established as an Empirical law, we may 
conclude that any average A is B, with a probability 
proportionate to the preponderance of affirmative 
over negative instances in our experience, care being 
taken that our A is a fair average instance. 

2. Application of two or more approximate generalisa- 

tions to the same case. This may occur in two 
ways : — • 

(a.) By addition of probabilities, — the " Self-cor- 
roborative chain" of Bentham. The type is this — 
" Most A is B," " Most C is £," this case is both A 
and 0,.\ it is probably B. 

Thus, Most of M's assertions are true (two in 
three) ; Most of JY's assertions are true (three in 
four) ; this assertion is both J/'s and N's, — what is 
the chance that the assertion is true in which they 
both thus concur ? If we arrange twelve statements 
of each, one under the other, we shall see that they 
both together speak truth six times in twelve, and 
both false together once in twelve, — therefore, if 
they both agree in an assertion, it will be true six 
times for once it is false, or the chances of its truth 
are ']. 
(b.) By Deduction, — "Self-infirmativc chain" of Ben- 
tham. The type is " Most A is 5," " Most C is jA 
this is a C, .'. it is probably an A, ■■■ it is probably 
a B. 

Here the degree of probability of the inference 



APPROXIMATE GENERALISATIONS. 187 

decreases at every step, and even where two proba- 
bilities only are concerned, may amount almost to 
nothing. But if the probabilities are taken fairly in 
reference to each other, we may say that the proba- 
bility arising from the two propositions taken in 
this way together is equal to the probability arising 
from the one abated in the ratio of that arising from 
the other. 

Thus — Most inhabitants of Stockholm are Swedes 
(eight out of nine) ; most Swedes have light hair 
(nine out of ten) ; .\ probability that any given in- 
habitant of Stockholm is light haired is -^ (=§ X jo)- 
The first form (a.) is exemplified when we prove a fact by 
the testimony of two or more independent wit- 
nesses ; the second (&.), when we adduce the testi- 
mony of one witness that he has heard the assertion 
from another. 

VI. There are two cases in which reasonings, 
depending upon approximate generalisations, 
may be carried as far, and are as strictly valid, 
as where based upon universals. 

This is an example of the current saying, " An exception 
which proves the rule ;" approximate generalisations 
are capable of being made use of in this way, be- 
cause they can be turned into general propositions 
in form or in fact. 

1. First Case. When the approximate generalisations are 
of the second kind — i.e., when we are cognizant of 
the character which distinguishes the cases which 
accord with the generalisation from those which are 
exceptions to it. In such cases we can, if we choose, 
substitute a universal proposition with a proviso, 



188 APPROXIMATE GENERALISATIONS. 

for the approximate generalisation ; and however 
many steps are involved in a train of reasoning, the 
successive provisos being carried forward to the 
conclusion will exactly indicate how far that conclu- 
sion may be positively relied on. 

Thus, take these approximate assertions, — " Most 
persons possessing uncontrolled power employ it 
ill," and "Most absolute monarchs possess uncon- 
trolled power,". 1 . "Most absolute monarchs employ 
their power ill ;" we may change them respectively 
into universals thus — " All persons not of unusual 
strength of mind and confirmed virtue possessing, 
&c. ;" " All absolute monarchs not needing the active 
support of their subjects possess, &c." ; .•. " All ab- 
solute monarchs not of unusual strength of mind, 
<fec, and not needing, &c, employ their power 
ill." 
Second Case is where the inquiry relates not to the pro- 
perties of individuals, but of multitudes ; since what 
is true approximately of the individuals, is true 
absolutely of the mass. 

For example — " Most Englishmen are Protestants" 
is equivalent to " The English people are a Pro- 
testant people." 

This is chiefly exemplified in the groups of Social 
Sciences ; and hence we see the error in the common 
opinion that speculations on Society and Govern- 
ment, as resting on apparently mere probable evi- 
dence, must be of inferior certainty and reliability. 



REMAINING LAWS OF NATURE. 189 

CHAPTER XXIV. 

Remaining Laws of Nature. 

We have already seen (pp. 30 and 31) that every 
Real Proposition must assert one of the following : — 

Existence, 
Order in Time, 
Order in Place, 
Resemblance. 

Up to this point we have considered the Logic of 
Propositions which assert Order in Time (including 
Causation and the other forms of Sequence and Co- 
existence in Time), the modes in which they are 
proved, and the conditions of their validity. It now 
remains to do the same for each of the other three 
great classes into which assertions may be divided. 

I. Propositions asserting Simple Existence. 

a. The existence of a phenomenon is but another 
word for its being perceived, or for the inferred 
possibility of perceiving it by somebody or other. 



It is true that a thing is often said to exist when it is 
absent or past, and is not and cannot be perceived ; 



i 



190 REMAINING LAWS OF NATURE. 

but even then its existence is to us only another 
word for our conviction that we should perceive it 
upon a certain supposition, — if we were placed in 
the needful circumstances of time and place, and 
endowed with the necessary perfection of sensory 



b. Existence of anything is established either — 

1. By immediate observation, when the thing is within 

the range of immediate observation ; or, 

2. By inferring its existence through marks or evidences, 

when it is beyond that range; i.e., from other phen- 
omena known by Induction, to be connected with 
the phenomenon by way of succession or coexist- 
ence. In such cases we prove the existence of a 
thing by proving that it follows, precedes, or coexists 
with some known thing, and this last is accom- 
plished by ordinary Inductive processes. 

c. General propositions affirming Simple Existence 
are sufficiently proved by a single instance. 

That "ghosts" or "sea serpents" exist would be posi- 
tively established if it could be shown that such 
things had even once been really seen. 
£It should have been remarked as regards the ordinary 
Inductive methods, proving coexistence or sequence, 
that two instances at least are necessarily required, 
though I Ik we two instances may he given by a single ex- 
periment; and it is this that is really meant when 
we speak of a single instance being sufficient to sup- 
port such a generalisation. If we refer to the canon of 
the Method of Difference (which applies to such 
cases) we shall see that two instances are necessary, 



REMAINING LAWS OF NATURE. 191 

— one of the presence, the other of the absence of 
the phenomenon. It is only, therefore, generalisa- 
tions asserting existence, which are really established 
by a single instance."^ 

II. Propositions asserting Resemblance (or 
the contrary). 

(Resemblance and its opposite include likeness, similarly ; 
and of number or magnitude — equality, inequality, 
similarity, proportionality.) 

Resemblance between two things may be established 
in two ways : — 

1. Immediately — by direct comparison of the two things. 

2. Mediately — when each is compared with some third 

thing. 

The comparison of two things through the inter- 
• vention of some third thing, when their direct com- 
parison is not practicable, is the appropriate scien- 
tific process for ascertaining Resemblances, and is 
the sum total of what Logic has to teach on the 
subject. 

[Error of Locke and Condillac school. 

This error consisted in the view that Reasoning itself 
was nothing but the comparison of two ideas 
through the medium of a third ; and knowledge, the 
perception of the agreement or disagreement of two 
ideas. 

This view of Reasoning and Knowledge requires 
to be restricted to our Reasonings about and our 
Knowledge of Resemblances, and not even then is 
the comparison made between our ideas of pheno- 



i 



192 REMAINING LAWS OF NATURE. 

mena but between the phenomena themselves. It 
is only in mathematics that the comparison is really- 
made between the ideas of things, and it is so be- 
cause our mental pictures of form, of lines, circles, 
&c, are as much fitted for comparison as pictures 
of them upon a surface before us, — being perfect 
transcripts, qua form, of the realities.] 

Propositions asserting Kesemblance may be divided 
into two classes : — 

1. Where the resemblance arises merely from a cause 

operating in a certain way. Thus, the angle of inci- 
dence equals the angle of reflection, asserts resem- 
blance, but it is only in consequence of being the 
very law of the cause to act in that manner. 

2. Resemblances true of all phenomena without refer- 

ence to their origin. These are the Laws of Num- 
ber and Extension, in other words of Mathematics. 
The first class of resemblances are merely part of the 
laws of the production of the phenomena, and these 
are amenable to the principles of ordinary Induc- 
tion. It is only the latter class which it is necessary 
to consider here, and we may therefore say that, of 
propositions asserting Resemblance, the laws of Mathe- 
matics alone require a separate logical consideration. 
A similar assertion we shall also find to be true of 
the next class of propositions, those asserting Order 
in Place; which will therefore be next dismissed, 
leaving only the Mathematical Sciences generally 
for final consideration. 

III. Propositions asserting Order in Place 

May be classed thus : — 
1. Those asserting Order in Place of the effects of causes, 



l 



METHODS OF PUEE MATHEMATICAL SCIENCES. 193 

are mere corollaries of the laws of the causes, and 
resolved by ordinary inductive processes. 

2. Those asserting the Order in Place or the collocation 

of the primeval causes. These assert in each in- 
stance an ultimate fact, in which no laws or uni- 
formities are traceable. 

3. The only remaining general propositions asserting 

Order in Place are some of the propositions of 
Geometry; laws through which we are able, from 
the situation (Order in Place) of certain points, 
lines, or areas, to infer that of others which are con- 
nected in some assigned manner with the former, 



THE METHODS OF THE PURE MATHEMATICAL 
SCIENCES. 



I. These are : — 

Science of Number, 
including 



Arithmetic, 

Algebra, 

The Calculus, &c. 



Science of Extension ) _, 

> Geometry, 

or Space, ) J 



II. General remarks on the fundamental 
principles of Mathematics : — 

1. These fundamental principles are (1) Axioms, and (2) 
Definitions so-called ; the last involving a postulate 
or implied assertion of the real existence of the 
thing corresponding to the name denned, which 

N 



194 METHODS OF PURE MATHEMATICAL SCIENCES. 

postulate can alone form the basis or premiss of 
scientific Deduction. 

2. These fundamental principles are based upon experi- 

ence ; being, in fact, proved by an Induction per 
enumerationem simplicem, and one of such wide 
generality as to be a perfectly rigorous proof. 

3. Comparison of the ideas of things (as regards their 

form) is strictly equivalent to comparison of the 
things themselves, since our mental pictures of not 
too complex forms are perfect transcripts of the 
realities. 

4. The following include the reasons why the primary 

truths of Mathematics seem to have a greater cer- 
tainty than other inductive truths : — 

a. Their universality. — They are true of everything, 

everywhere, and at every time. 

b. Tlieir extreme familiarity. — The perception of their 

truth only requires the simple act of looking at 
objects in the proper position, and often only 
thinking of them in such a position. Hence 
exemplifications of their truth are incessantly 
presented to us. 

c. The absence of any analogies to suggest a different 

law. — This is very important; if everything in 
the universe always maintained a condition of 
absolute rest, we might find it as difficult to 
conceive the possibility of the sun falling from 
the sky as we now have of conceiving that two 
straight lines can enclose a space. 

d. Tl>'}) are never counteracted, being independent of 

causes. 



THE SCIENCE OF NUMBER. 195 



III. The Science of Number. 

1. Generally. 

a. The elementary or fundamental truths of this branch 

of Mathematics are : — 

/ Things equal to the same are 

_ . i equal to one another. 

1. Two axioms — < T * i i jj j x i 

\ if equals be added to equals 

V the sums are equal. 

{Of the various numbers, &c, 
explaining a name and assert- 
a fact. 

The other axioms may be deduced from these two. 

b. Every name of a number (one, two, three, &c.) denotes 

physical phenomena, and connotes a physical pro- 
perty of those phenomena, and that is the charac- 
teristic manner in which the agglomeration or whole 
is made up and may be separated into its parts ; in 
other words, the manner in which objects must be 
put together to form that number. Thus the name 
"two" connotes an impression on the senses similar 
to that made by . . ; "three" by . . . or .•., and 
so on ; the higher the number the more the ways in 
which it may be made up. 

c. All the theorems of the Science of Number assert the 

identity of different modes of formation. They assert 
that some mode of formation from x, and some 
mode of formation from a given function of x, pro- 
duce the same result. 

d. The general problem of the Science of Number is — Given 

a function, what function is it of some other func- 
tion 1 F (= 3 a 2 ) being a certain function of a given 



196 THE SCIENCE OF NUMBER. 

number (a), what function will F be of any function 
of that given number (a) 1 

(A function of x being any expression which con- 
tains x.) 

2. Arithmetic. 

a. Every arithmetical proposition, every statement of 

the result of an arithmetical operation, is the state- 
ment of one of the modes of the formation of a given 
number. 

Suppose we take the number 1728, — this may be 
considered as made up in an infinity of ways ; thus, 
as 1000 + 700 + 20 + 8, &c. &c. To say 12 3 = 1728 
is to affirm that one way of making up 1728 is to 
cube the number 12. When one mode of forming a 
number is given we can find any others which are 
required. 

b. What renders Arithmetic a Deductive Science is the 

applicability to it of the comprehensive law — " The 
sums of equals are equals," or " Whatever is made 
up of parts is made up of the parts of those parts." 
Every arithmetical operation is an application of 
this law or some law deducible from it. 

3. Algebra. 

a. The propositions of Algebra affirm the equivalence of 
different modes of formation of numbers generally ; 
they are true not only of particular numbers, but 
of all numbers. 

It is true that different numbers cannot be formed! 
in the same way from the same numbers, but they| 
may be formed in the same way from different num- 
bers ; and this way algebraical propositions assert. 

J>. Algebraical Notation is a system of nomenclature de-j 



SCIENCE OF EXTENSION. 197 

vised for the purpose of enabling us to carry on 
general reasoning about functions, by enabling us 
to express any numbers by names which, without 
specifying what particular numbers they are, shall 
show what function each is of the other ; in other 
words, shall show the mode of formation from one 
another. 
The general problem of Algebra is — F being a certain 
function of a given number, to find what function 
F will be of any function of that given number ; or, 
given a function, what function is it of some other 
function 1 



IV. Science of Extension (Geometry). 

a. The elementary principles of this branch of science 

are, as before explained, Axioms and Definitions, with 
their contained postulates. 

b. Every theorem of Geometry is a law of External 

Nature ; and this would have been perceived (or 
that Geometry is a strictly physical science), in all 
ages, had it not been for the illusion produced by 
two causes. 
(1.) The characteristic property of the fundamental 
facts, that they may be collected from our ideas 
or mental pictures of the objects as effectually 
as from the objects themselves. 
(2.) The demonstrative character of geometrical truths £ 
i.e., certain suppositions or hypotheses being 
granted, Geometry deduces from these what 
conclusions it can. 

c. Why Geometry is so eminently deductive. 

(1.) All questions of position and figure can be resolved 



198 SCIENCE OF EXTENSION. 

into questions of magnitude ; * and thus Geo- 
metry is reduced to the single problem of the 
ascertainment of the relations of quantity (chiefly 
equality;, sometimes proportionality) between 
magnitudes ; and by the aid of the axioms re- 
lating to equality, each equality becomes a mark 
of an almost infinite number of others. 
(2.) Three of the principal laws of extension or space 
are unusually fitted for rendering one position 
or magnitude a mark of another, and thus mak- 
ing the science deductive. These are : —(a.) Mag- 
nitudes of enclosed spaces are determined by 
magnitudes of enclosing sides and angles ; (b.) 
the length of any line may be measured by the 
angle it subtends, and vice versa; and (c.) the 
angle which two straight lines make with each 
other at an inaccessible point is measured by 
the angle which they severally make with any 
straight line we choose to select. 

V. Function of Mathematics in other sciences. 

Causes, like everything else which can have quantity or 
position, operate pro tanto under mathematical laws ; 
and in proportion as any science affords precise 
numerical data, so is the applicability of mathe- 
matical principles to that science. 

VI. Limits of that function. 

(1.) When causes are so imperfectly accessible that we 
cannot get their numerical laws. 

* Thus the position and figure of any line, plane, or solid, is deter- 
mined if wo know the position of a sufficient number of points in it; 
and the position of any point may bo fixed by the magnitudes of two 
or tbree co-ordinates drawn in reference to two or three uxes. 



ON THE GROUNDS OF DISBELIEF. 199 

(2.) Where they are so numerous or complexly inter- 
mixed that the calculation would transcend human 
powers. 

(3.) Where the causes are perpetually fluctuating, as in 
Biology and Sociology. 

The value of Mathematics as a preparatory study is 
chiefly as presenting the most perfect type of the 
Deductive Method. 



CHAPTER XXV. 

On the Grounds of Disbelief. 

By Disbelief is here meant not mere doubt, but positive dis- 
belief ; so that even if evidence of apparently great strength 
is, or may be, produced for the proposition, we consider the 
witnesses in error, or the evidence somehow wrong. 

The positive evidence produced in favour of an assertion 
that is thus disbelieved is always grounded on some approxi- 
mate generalisation; such as — "Most things asserted by a 
number of respectable witnesses are true ; " or " Most of the 
impressions made on the senses accord with reality." The 
affirmative evidence, then, being never more than approxi- 
mate generalisation, the whole question depends upon what 
the evidence against the proposition is. If this is an approxi- 
mate generalisation, it is a case of comparison of probabilities ; 
but if the evidence be a higher truth, we are guided by the 
following : — 

I. Canon of Disbelief. 

If an assertion be in contradiction (not 



200 ON THE GROUNDS OF DISBELIEF. 

merely to any number of approximate gener- 
alisations, but) to a complete generalisation, 
grounded on a rigorous induction, — the absence 
of adequate counteracting causes being sup- 
posed, — it is impossible, and is to be disbe- 
lieved totally. 

An induction of the kind referred to may be obtained 
when the Inductive Methods capable of giving 
rigorous results have been rigorously carried out. 
If, indeed, we rind it necessary to admit the in- 
consistent assertion, we must give up the law, as 
somehow involving a mistake. 

II. Two cases may be considered under this 
Canon : — ■ 

1. Where the assertion appears to conflict with a 
real law of Causation. 

Facts contrary to experience (Hume). Facts disconformable 
in toto or in genere (Bentham). 

This is probable or improbable in exact proportion to 
the probability or improbability that there existed 
in the case in question an adequate counteracting 
cause. 

2. Where the assertion conflicts with mere uni- 
formities of coexistence, not proved to he dependent 
on Causation. 

Facts unconformable to experience (Hume). DisconformaWy 
fa *]>ccic (l)cntham). 



ON THE GROUNDS OF DISBELIEF. 201 

It is with these uniformities principally that the mar- 
vellous stories of travellers conflict. What is really 
asserted in cases of this nature is the existence of a 
new kind. This is not by itself at all incredible, 
and having fairly considered the probability that 
the alleged new kind could have escaped previous 
observers, the assertion is to be tested by the prin- 
ciple before laid down, — that it is improbable in 
proportion to the generality of the uniformity with 
which it conflicts. 

[III. Hume's doctrine of Miracles. 

Hume, as is well known, laid down an argument against 
miracles in this form, — " It is improbable that a 
miracle should be true ; it is improbable that the 
testimony in its favour should be false ; we have 
therefore a comparison of improbabilities ; but the 
former improbability is the highest possible, being 
an impossibility, for ' anything which is contrary to 
experience is impossible.' " 

The last proposition is the important part of the argu- 
ment, and it is, in fact, nothing more than a loose 
statement of the Canon of Disbelief. But on 
referring to that Canon, we find that the incredible 
assertion must not be merely that a cause existed 
without being followed by its effect, but that this 
happened in the absence of counteracting causes. 

Now the assertion in the case of an alleged miracle is 
the exact opposite of this, — it is, that the effect was 
defeated, not in the absence, but in consequence, of a 
counteracting cause ; that cause being the direct 
action of supernatural power. If such a cause exist 
in the case, there is no question of its competency ; 



202 ON THE GROUNDS OF DISBELIEF. 

the only improbability is whether it really did 
operate in the given instance. 
All, therefore, that Hume has made out is this — that no 
evidence can prove a miracle to any one who does 
not previously believe in the existence of a superior 
being, or who thinks that such being would not 
interfere. Nor, again, is it possible for the miracle 
itself to prove the existence of supernatural agency, 
for there is always the alternative possibility of some 
unknown natural cause.] 

IV. The absolutely impossible assertions 
Are such as contradict the Laws of Number 

or Extension, or the universal law of Causation. 

An assertion at variance with either of these is 

absolutely and for ever incredible, — in other 

words, impossible. 

Summary. — We have thus far, then, considered 
assertions conflicting 

(1.) With approximate generalisations — we must balance 

probabilities. 
(2.) With laws of causation — probability depends on 

probability of presence of counteracting cause. 
(3.) With uniformities of coexistence — upon probability 

of existence of the new kind. 
(4.) With laws of Number, Extension, or Causation, — 

absolutely impossible. 

And wc now proceed to consider one or two minor 
points. 



ON THE GKOUNDS OF DISBELIEF. 203 

V. Improbability before the fact (i.e., the 
chances being against a thing before it hap- 
pens) must not be confounded with improba- 
bility after (difficulty of believing a thing said 
to have actually occurred). 

Many events are altogether improbable to us before they 
have happened, or before we are informed of their 
happening, which are not in the least incredible 
when we are informed of them, because not con- 
trary to any, even an approximate, generalisation. 
That a given individual will die in a particular man- 
ner, at a particular moment, twenty years previously 
is highly improbable, though we believe the state- 
ment directly we are assured it has happened on 
credible authority. Dr Campbell in his reply to 
Hume has confounded these two kinds of improba- 
bility. 

VI. Probability of Coincidences. 

Suppose a person assures us that he has seen the 
six thrown ten times in succession by a die, 
which we have by previous trials ascertained to 
be perfectly fair, what is the credibility of the 
assertion 1 

It is perfectly evident, in the first place, that a series of 
ten sixes is, in itself, just as probable as any other 
series of ten numbers, — if therefore we are inclined 
to discredit the assertion of the former, where we 
should believe the latter, it must be, not because the 
former assertion is less likely to be true, but because it 
is more likely to be false. Motives to falsehood, of 



204 ON THE GROUNDS OF DISBELIEF. 

which one of the most frequent is the desire to 
astonish, are more likely to have operated in the 
wonderful assertion. 
Such a regular series certainly seems more improbable 
than any given irregular series, but this is only be- 
cause the comparison is tacitly made between it and 
all irregular series taken together. Of course if 
before the ten throws we were asked to guess whether 
the series would be regular or irregular, we should 
say irregular, but this is only because the possible 
irregular combinations are immensely more nume- 
rous ; if we had to determine whether a given irre- 
gular or a given regular series were more probable, 
it would be quite indifferent on which we fixed. 



BOOK IV. 



OPERATIONS SUBSIDIARY TO INDUCTION. 



CHAPTER I. 

Observation and Description. 

The " operations subsidiary to Induction" are Observation, 
Abstraction, Naming, and Classification. Observation is here 
considered ; and the term in this place is understood as in- 
cluding Experiment, not as contrasted with it. 

I. Observation takes its place first amongst 
operations subsidiary to Induction. 

For Induction is the extension of something which has 
been observed to be true in certain members of a 
class to the whole class. As far as Logic is con- 
cerned (its function being the estimation of evi- 
dence), we have only to consider in reference to 
observation, — 



206 OBSERVATION AND DESCRIPTION. 

II. What condition is necessary in order 
that any fact supposed to be observed, may 
safely be received as true % 

The sole condition is that which is supposed to have 
been observed shall really have been observed ; 
that it is an observation, not an inference. 

We have already pointed out (p. 3), the large share occu- 
pied by rapid and unconscious inferences in what 
are ordinarily supposed to be direct impressions on 
the senses. In strictness, observation only extends 
to the sensations themselves which we receive from 
objects, or to the mind's internal states ; the very 
existence of the object is an inference from these 
sensations ; and what Logic, therefore, has to teacli 
on the point is this, — that, in such cases, we should 
be aware of what is really observed, and what is 
inferred, and should remember that an error may 
lurk in the latter process. Errors of sense so called, 
are, in fact, erroneous inferences from sense, the 
sensations themselves must be real. 

III. A description of an observation affirms 
more than is contained in the observation ; it 
is inherent in a description to be a statement 
of resemblance or resemblances. 

To describe an object is to assign attributes to it, and 
in doing this wo necessarily assert a resemblance 
between the object and everything else which pos- 
sesses any of those attributes. If I say, "This 
object is white," I incidentally affirm a resemblance 
between this object and all other white objects. 



ABSTRACTION. 207 

This resemblance to some other objects may be ascer- 
tained by direct comparison, as in the instance just 
given, or deductively by marks of the attribute. 
Thus, when we say, " The earth is a sphere," we 
affirm resemblance to all other spherical bodies ; 
but this is not obtained by direct comparison, but 
by marks of the spherical form in the case of the 
earth (these marks being circular horizon, disappear- 
ance of hull of ships first, &c.) 



CHAPTEK II. 

Abstraction, or the formation of Concepts. 

I. A general notion or concept is the con- 
ception of a multitude of resembling indi- 
viduals as an aggregate or class. 

Such general notions certainly exist in the mind, i.e., 
the mind can in some way or other conceive a mul- 
titude of individuals, as an assemblage or class ; 
otherwise we could not use general names with any 
consciousness of meaning. 

How such notions are obtained, and what is their pre- 
cise nature, are questions with which Logic is not 
concerned ; it is sufficient for its purpose that the 
name of a class calls up some idea by which we can 
think and speak of a class as such. 

II. A general conception of the phenomena 



208 ABSTRACTION. 

we are investigating is a necessary preliminary 
to that comparison which Induction presup- 
poses. 

That is, we cannot ascertain general truths, — truths 
applicable to classes, — unless we first form the 
classes ; and this is done by comparing individuals 
and ascertaining their common properties. 

III. General conceptions do not develop 
themselves from within the mind, but are 
always, in the first instance, impressed upon 
it from without. 

Dr Whewell holds that the mind is, as it were, a store- 
house of conceptions of every kind, which are pro- 
duced from it from time to time as they are found 
to suit particular aggregates of objects ; the mind, 
as it were, produces conceptions as a tree fruit, 
without assistance from impressions from outward 
things. Mill replies, that general conceptions are 
always obtained by abstraction from individual ob- 
jects, either — (1.) from the very things we are at the 
moment examining, of which we are endeavouring 
to ascertain the points of agreement ; or (2.) from 
things which we have perceived or thought of on 
some former occasion, whose points of agreement 
we have already ascertained, and which we find to 
coincide more or less accurately with the points of 
agreement of the phenomena we are at the moment 
investigating. Thus, suppose I have placed before 
me a number of points placed along the circumfer- 
ence of a circle ; if I have never seen a circle before, 



ABSTRACTION. 209 

I may gather a conception of it from the very in- 
stance before me ; if I have already the notion of a 
circle, I simply have to compare it with the present 
case, to find whether it agrees or not. 
It is evidently only in the latter class of cases that the 
conception can be said to pre-exist in the mind ; but 
even here the conception was originally obtained by 
observation and comparison of external phenomena 

IV. Type function of general conceptions. 

A general conception itself, originally the result of com- 
parison, becomes the type of comparison. 

The human mind cannot properly compare more than 
two objects together simultaneously. Having, by 
this first comparison, noted the most prominent 
points of agreement, or apparent agreement, and 
having thus framed our first rough general notion, 
when another object, apparently of the same class, is 
presented to us, we naturally compare it, not with 
either of the two first, but with the aggregate of 
points in which they agreed, — in other words, with 
our first general conception. This original concep- 
tion may thereby be found to require correction or 
to admit of extension ; and so we go on, comparing 
our conception with one individual after another, 
till we get a sufficiently precise general idea of the 
class. In a word, it is perfectly clear that when we 
have to compare a large number of individuals 
together, a type of some kind is indispensable, and 
the general conception is the only really useful type. 

V. Tentative process in forming or applying 
neral conceptions. 



i 



general conceptions 



210 ABSTRACTION. 

If, after forming our first rough general conception, we 
proceed further to compare it with one individual 
after another, if it do not seem suitable it becomes 
necessary either to correct it or even abandon it 
altogether. In the latter case we must begin again, 
and look out for a fresh set of agreements, and form 
and try a new conception. 

It is this tentative process which seems to have suggested 
Dr "Whewell's view — that general conceptions are 
furnished by the mind and from the mind itself. 
In such cases, it is true, the conception is furnished 
by the mind, but not till it has first been furnished 
to the mind by the contemplation of outward phen- 
omena. 



VI. What is meant by appropriate concep- 
tions. 

Appropriate is an elliptical term, meaning appropriate 
to some particular purpose ; and an " appropriate 
conception " is one which comprehends not only real 
points of agreement, but also such points of agree- 
ment as are important relatively to our particular 
purpose. 

Thus, a gardener's conception of u seed," " fruit,'' or 
" flower," would be different from a botanist's, yet 
each would be most appropriate for its own purpose. 

VII. What is meant by clear conceptions. 

A conception is clear when we know exactly in what 
the agreement between the different phenomena 
consists ; in other words, when we know accurately 



ABSTRACTION. 211 

what simpler notions our general conception is made 
up of, or comprehends. 
It is not necessary for a conception to be clear that it 
should be complete,— that is, that we should know 
all the common properties of the things we class 
together. 



VIII. The appropriateness of our concep- 
tions (or rather the chance of our hitting upon 
points of agreement appropriate to our par- 
ticular purpose) depends chiefly upon the 
activity of our observing and comparing 
faculties. 

The clearness of our conceptions depends 
chiefly upon the carefulness and accuracy with 
which we employ those faculties. 

The chief requisites for clear conceptions are, therefore — 
(1.) Habits of attentive observation ; (2.) Extensive 
experience ; and (3.) A memory which receives an 
exact image of what is observed, and tenaciously 
retains it. 



I 



212 NAMING, AS SUBSIDIARY TO INDUCTION. 



CHAPTER III 

Naming, as subsidiary to Induction. 

Preliminary remarks. 

1. The remarks in this chapter relate to general 
names, that portion of language with which Logic is 
chiefly concerned. 

2. The logical use of names is either: — 

(1.) Indirect, — the same as their uses as instruments of 
thought. 

(2.) Direct, — enabling us to lay down general proposi- 
tions. 

3. The use of names as instruments of thought con- 
sists — 

(1.) In their power of binding up simpler ideas into 
convenient groups ; and (2.) enabling us to produce 
these groups when wanted. In these ways they 
form a powerful artificial memory, and very much 
shorten thinking. 

General names aggregate together attributes into such 
groups as iire wanted for use ; a name, for example, 
like " civilisation," binds up into one whole a num- 
ber of simpler ideas — a certain state of intellectual 
cultivation, a certain condition of moral feeling, and 
of the knowledge of art and scieuce, &c. If we had 



213 

no such word as " civilisation," we should not only 
be compelled to enumerate all these every time we 
wanted the complex idea, but we should probably 
fail to remember them, since there would be nothing 
to throw them into a group and rivet them to- 
gether. So also is it with other general names ; 
they, in fact, perform in the mind the same function 
as the binding does to the books of a library ; with- 
out such, the mind would resemble a library of books, 
all in separate leaves, confusedly mixed. 

I. Names are not absolutely indispensable 
for inference. 

We can make some inferences without the use of lan- 
guage, namely, simple cases of direct inference from 
particular cases to another particular case, without 
passing through a general proposition. If reasoning 
consists in recognising one phenomenon as a mark 
of another, nothing clearly is absolutely requisite, 
except senses to perceive that two phenomena are 
conjoined, and an associative law, by means of which 
one of these raises up the idea of the other. 

II. That our inferences without language, 
however, would be very limited and precarious 
may be inferred when we remember that the 
direct use of names is to enable us to lay down 
and preserve general propositions ; and that 
the uses of general propositions are : — 

1. They enable us to avail ourselves of our past experi- 
ence, by regularly registering it in assertions. 

2. As also of the collective experience of mankind. 



214 PRINCIPLES OF DEFINITION. 

3. They enable us to permanently record or register 

uniformities. 

4. They enable us, moreover, to do this once for all. 

III. General names are not a mere contriv- 
ance for economising language. 

Some have supposed that the necessity of general names 
springs from the immense multitude of individual 
objects, preventing us having a separate name for 
each of them of which we may wish to speak. This, 
if true to some extent as regards the use of general 
names in ordinary language, is not true of their 
logical use. Even if we had a separate proper name 
for each individual thing, without general names we 
could register neither the results of our comparisons 
nor uniformities. 

Rigorously speaking, we could carry on logical opera- 
tions without any other general names than the 
abstract name of attributes ; our propositions haviDg 
this form, "A possesses such and such an attribute ; " 
or this — "Attribute A is conjoined with or re- 
sembles attribute B." 



CHAPTER IV. 

I'KixcirLES of Definition. 

The chief requisites of a ■pliiloso'pliical Ian- 
guage (i.e., a language perfectly suitable for 



PEINCIPLES OF DEFINITION. 215 

the investigation and expression of general 
truths) are : — 

1. That every general name employed should have a 

meaning precisely determined and steadily fixed. 

2. That we should possess a name wherever one is 

needed ; and that this name should fulfil certain 
ends in the best manner. 
It is to the discussion of the first of these, or the Prin- 
ciples of Definition, that this chapter is limited. 

I. To attach a certain and definite meaning 
to every general name, is the same thing as 
to assign to every general concrete name a 
definite connotation ; for if the name be ab- 
stract, its meaning is at once settled by the 
connotation of the corresponding concrete. 

II. Concrete general names are very often 
used with indefinite connotation ; and this 
arises in two ways : — 

1. Such words are used without connoting any distinct 

attribute at all, but merely a vague general resem- 
blance to other things called by the same name. 

When ordinary people speak, for instance, of an 
action as "just" or " noble," they imply no distinct 
quality of the act, but merely that it resembles acts 
to which they have been accustomed to hear those 
terms applied. 

2. The connotation derived from accustomed predication 



216 PRINCIPLES OF DEFINITION. 

often assists in giving a vague meaning to the 
name. 

When a general name stands as the subject of a 
proposition, predicates or attributes are affirmed of 
the objects it stands for. Now, if a name is fre- 
quently employed as a subject, and a certain set of 
attributes are often predicated of it, these attri- 
butes often become mixed up in a vague way with 
the original meaning of the name. Just as from 
frequently hearing the ass called stubborn or foolish, 
we come to regard the name "ass" as connoting 
stubbornness or foolishness. 

III. The framing of a satisfactory definition 
of a name in common nse is not an arbitrary 
process ; neither does it depend wholly upon 
such common usage, but also upon a knowledge 
of the properties of the things denoted. 

For, in order to assign a connotation to a name, consist- 
ently with its continuing to denote certain objects, 
we have to make our selection from amongst those 
attributes in which the objects agree. The process 
involves two inquiries — In what do the objects 
agree ? What attributes have they in common ? 
And, having settled this, the second is — Which of 
these common attributes will serve best to mark 
out the class from all others, and ought therefore 
to be assigned as the connotation of the name ? 

In answer to this second inquiry, we say, — those 
attributes ought to be selected which are, as far as 
possible — 



PRINCIPLES OF DEFINITION. 217 

1- Sure marks of the greatest number of other important 
properties. 

2. Such as are familiarly predicated of the objects. 

3. Such as have the greatest share in producing the 

general resemblance amongst the objects. 

IV. The transitive application of words. 

1. What is meant by it. 

When men meet with an object new to them, there is a 
strong tendency not to invent a new name, but to 
apply to the new object the name of some familiar 
object which seems to resemble the new one most. 
Thus the word " oil " {oleum, oliva) originally meant 
olive oil exclusively, but as new objects were con- 
tinually being discovered bearing more or less re- 
semblance to olive oil, the name oil was by degrees 
extended to a very large number of bodies, — to sul- 
phuric acid, for example ; and even to solids, as 
palm oil. Now a name, in this transitive way, may 
pass on from object to object, till at last the deno- 
tation becomes so wide that the various included 
objects have but little or anything in common. 

2. How the logician should deal with such cases. 

He should be careful not unadvisedly to discard any of 
the connotation of the name, but should, if it be 
necessary, rather restrict its denotation, by dropping 
some of the objects to which it has been extended. 

3. Important law of mind in reference to these 
transitions. 

When the word has passed naturally and easily from one 



218 

shade of meaning to another, the association between 
the different meanings may become virtually indis- 
soluble, and the various transitive meanings will 
coalesce into one complex conception ; the meanings 
will blend together in the mind, and the real tran- 
sition becomes an apparent generalisation. 

V. It is an important fact that there is a 
constant tendency in names to lose portions 
of their connotation, from habitual inatten- 
tion to the total of ideas conveyed by the 
name ; and this is especially likely to occur 
when the connotation is left vague and un- 
settled. 



CHAPTER V. 

History of Variations in Meanings of Terms. 

I. Accidental connotation ; collateral asso- 
ciations affecting words. 

The incorporation into the meaning of a word of some 
circumstance originally accidental is a frequent 
cause of variation in meaning; and such accidental 
connotation may not only be incorporated into the 
word, but may in the end more or less completely 
supersede the original meaning. 

A name which is in every one's mouth derives its con- 



HISTOKY OF VARIATIONS, ETC. 219 

notation from the circumstances which are habitu- 
ally brought to mind when it is pronounced ; but if 
any circumstances happen to be so frequently asso- 
ciated with these as to be constantly suggested when 
the name is used, they may become as much part of 
the meaning of the name as those originally brought 
to mind by it. Thus, pagan originally meant 
"dweller in a village;" but since such persons were 
usually behind the age, ignorant and heathenish, 
these accidental circumstances gradually became in- 
corporated with the meaning of the name, and at 
length formed its exclusive meaning. 

This continual incorporation of meanings origin- 
ally accidental is the reason why — 

(1.) There are so few exact synonymes in a language ; and 
(2.) Why the dictionary meaning of a word is often so 
imperfect an exponent of its real meaning. 

II. Transitive change in meaning of words 
has been already noticed. (P. 217.) 

III. Alterations in the meaning of a term 
must evidently consist either in one of these 
two things, or of both together : — 

1. Loss of some part of connotation (Generalisation). 

2. Taking on of fresh connotation (Specialisation). 
That is, a word must either come to mean more or to 

mean less, or to mean more in one direction while 
meaning less in another. Of course as connotation 



220 HISTORY OF VARIATIONS, ETC. 

is increased, denotation is diminished, and vice 
versa. 

1. Generalisation. 

May happen : — 

(a.) From dropping a part of connotation from mere 
ignorance of the omitted portion. 

(b.) The fact that the number of known objects multi- 
plies faster than names for them gives rise to a 
tendency to give to a new object the name of an 
old ; and thus the name, extending its denotation, 
lessens its connotation. 

Is most likely to occur to the greatest extent in words 
expressive of the complicated phenomena of mind 
and society. 

2. Specialisation. 

May happen : — 

(a.) Words originally expressing a general character be- 
coming limited to some particular object. 

Thus Arsenicum meant originally any strongly 
irritant substance, but afterwards became limited 
to the substance now known by that name. 

(b.) From the habit which persons, whose attention is 

frequently directed to certain species of a genus, 

have of giving the name of the genus to that species. 

Thus, to a sportsman u bird " means a " partridge " 

or "grouse." 

(c.) An idea sometimes becomes incorporated into the 

meaning of a word from mere chance conjunction. 

(See Accidental Connotation.) 

I From the common practice of using general terms 

where more specific words might be employed ; 



TERMINOLOGY AND NOMENCLATURE. 221 

thus the wider term gradually gets a specific con- 
notation. 

A practice has a tendency to grow up in a polite 
society of designating objects by the most general 
words which will suffice to point them out ; and 
thus such words often pick up additional meanings. 
The additional connotation which a word soonest and 
most readily takes up is that of agreeableness or 
disagreeableness in some of its forms ; that a thing 
is good or bad, desirable or the reverse, and so on. 



CHAPTER VI. 
Terminology and Nomenclature. 

Having in Chapter IV. discussed the first of the two main 
requisites of a philosophical language, we now in this chapter 
proceed to the consideration of the second of these, namely : — 

I. That we should possess a name wherever 
one is needed (i.e., there should be a name for 
everything about which we have often to 
speak). 

[For the subordinate clause — " this name 
should fulfil certain ends in the best manner," 
—see Chap. VII., p. 208.] 

This second requisite involves three sub-re- 
quisites : — ■ 

1. An accurate descriptive Terminology. 



222 TERMINOLOGY AND NOMENCLATURE. 

2. A name for each important result of Scientific Ab- 

straction. 

3. A Nomenclature, or System of Names of Kinds. 

1. An accurate descriptive Terminology. 

A Terminology is a System of Terms or Names ; and by 
a name being " accurately descriptive " is meant 
this — that it should be capable of conveying an 
exact notion of the phenomenon to another person, 
as do the words "hunger," "blackness," &c, in com- 
mon speech. 

Strictly speaking, a name for every variety of simple or 
elementary feeling would be sufficient ; but it con- 
duces much both to brevity and clearness to have 
separate names for oft-recurring combinations of 
feelings. 

When a name is appropriated to a previously unnamed 
phenomenon, the new name ought to be associated 
immediately with the phenomenon or feeling to 
which it has been assigned, so as to recall it with- 
out delay or trouble. 

2. A name for every important result of Abstrac- 
tion , — i.e., a name for every important common pro- 
perty, or aggregate of common properties, which we 
detect by comparison of the facts. 

There are three advantages connected with the 
appropriation of a single definite name to the ab- 
stracted quality : — 

(«.) Its use saves time, space, and circumlocution. 
(6.) It promotes perspicuity by enabling us to reason 
with or about the conception as a whole, without 






TERMINOLOGY AND NOMENCLATURE. 223 

being confused by thinking unnecessarily of its 
parts ; just as mathematicians substitute a single 
symbol for a complex expression. 
(c.) A name fixes our attention upon a phenomenon and 
causes it to be remembered. 

3. Tliere must be a name for every Real Kind, — 
in other words, a Nomenclature. 

A Nomenclature may be defined as — A collection of the 
names of all the Real Kinds with which any branch 
of knowledge is conversant ; or more strictly, Of 
all the lowest Kinds or Infimas Species. 

Such is exemplified in Botany, Zoology, and Chemistry : 
Viola Odorata, Felis leo, Ferric oxide, are examples 
from Systems of Nomenclature. 

There is a peculiarity in the connotation of Names which 
form part of a Nomenclature, — namely, that besides 
their ordinary connotation, as concrete general 
names, they have a special one ; besides connoting 
certain attributes, they also connote that those 
attributes are distinctive of a Eeal Kind. A defini- 
tion can only express the former ; and hence an 
appearance that the signification of such terms can- 
not be completely conveyed by a definition. 

II. A third and subordinate aphorism re- 
specting a philosophical language may be laid 
clown thus : — 

Whenever the reasoning can be carried on mechanically, 
without risk of error, the language should be ren- 
dered as mechanical as possible ; but if not, every 
precaution should be taken against such a mode of 



224 TERMINOLOGY AND NOMENCLATURE. 

using it. la connection with this we proceed to 
notice a few points respecting — 

III. Mechanical use of language. 

1. What is meant by it. 

The complete or extreme case is when language is used 
without any consciousness of meaning, and with 
only the consciousness of using certain visible or 
audible signs in conformity with technical rules 
previously laid down. 

2. When applicable and not. 

(a.) Mechanical use of language can never be permitted 
or be useful in our Inductive operations. 

(b.) And in our Deductive only when our reasonings are 
independent of any property peculiar to the things 
with which we are concerned, — i.e., only when we 
are concerned with properties which are properties 
of all things whatever. 

(c.) Therefore, practically speaking, its use is limited to 
our reasonings about Number. 

(d.) In all other sciences, then, except Number, we must 
endeavour as much as possible to prevent ourselves 
from using language mechanically ; and this end is 
accomplished (1.) by throwing as much meaning as 
possible into words ; and (2.) by frequently calling 
up the ideas involved in their meanings. 



CLASSIFICATION, ETC. 225 

CHAPTEE VII. 

Classification as subsidiary to Induction. 

There are two kinds of Classification (see also 
p. 37) :- 

1. That form which is inseparable from the use of gen- 

eral names. As has been already remarked, every 
name which connotes an attribute, incidentally 
divides all things into two classes, — those which 
possess, and those which do not possess, the attri- 
bute in question. Such a classification includes not 
only all things which are known or which exist, but 
all which may be imagined or hereafter be dis- 
covered. 

2. In the other kind of classification, with which alone 

we are here concerned, the arrangement or distribu- 
tion of things is not a mere incidental consequence, 
but the end and aim of the process ; the naming 
being secondary to and in conformity with the 
classification. Such are the classifications of Bo- 
tany, Zoology, &c. 

The principles of Scientific Classification have re- 
ference to a twofold object : — 

1. The arrangement of the objects of its study into 

Natural Groups, with the object of facilitating our 
inductive inquiries generally. 

2. The arrangement of Natural Groups into a Natural 

P 



226 CLASSIFICATION, ETC. 

Series, with the object of facilitating some special 
inductive inquiry. 
It must not be supposed that the arrangement into a 
Natural Series is merely a further stage of the 
arrangement into Natural Groups ; the two are dis- 
tinct both in their principle and their object. It is 
with the arrangement into Natural Groups that we 
are concerned in the present chapter ; the " Classi- 
fication by Series " is discussed in Chapter VIII. 

I. "Natural Groups" are classes of such 
a kind that the things included therein re- 
semble each other most in the general aggre- 
gate of their properties. Such groups of 
individuals, species, or genera, as would be 
spontaneously framed by any one acquainted 
with the whole of the properties of the things, 
but not specially interested in any. 

The object of a classification into such groups is the best 
possible ordering of our ideas in reference to the 
things, or to make us think of those objects together 
which have the greatest number of important com- 
mon properties. 

Its general problem is to provide that the things be 
thought of in such groups, and those groups in such 
an order, as will best conduce to the ascertainment 
and remembrance of their laws. 

II. General Principles which should govern 
the formation of a natural classification. 



CLASSIFICATION, ETC. 227 

All Eeal Kinds are Natural Groups ; and a Natural 
Classification must incorporate into itself all dis- 
tinctions of Eeal Kind in the objects with which it 
is concerned. 

The Infinise Species — the lowest classes, — in a Natural 
Classification, will (almost invariably) be the logical 
infimse species, — that is, the lowest Eeal Kinds. 

The next step is to class these into higher groups. Cer- 
tain species are, in the first instance, suggested to 
us by a feeling of general resemblance (i.e., by type) 
as being allied ; we then determine what characters 
these resembling species have in common, and by 
means of some of these we constitute our genus; 
and so on with the still higher groups. It is not, 
however, absolutely essential that all the characters 
assigned to the higher groups should be found in 
every lower group contained therein ; it is sufficient 
if any lower group contains enough of them to cause 
it to resemble the members of that higher group 
more than of any other. 

Upon what principle ought ive thus to select cha- 
racters for forming our groups f 

We must select such characters as will constitute our 
groups, so that the members thereof shall possess 
the greatest number and the most important of their 
properties in common. 

To this end one or both of the following requisites 
must be fulfilled ; and in proportion as they are ful- 
filled is the excellence of the classification : — 

(1.) The selected characters must themselves be im- 
portant properties. 



228 CLASSIFICATION, ETC. 

(2.) They must be marks of other properties, numerous 
and important. 

Therefore, if we can, we should select as our distinctive 
characters the causes of many other properties, be- 
cause — (1.) they are themselves important, and (2.) 
are the surest of marks. Again, properties upon 
'which the general aspect depends should, cat. par., 
be selected ; this, however, is not a sine qua 
non. 

III. How the names of Natural Groups 
should be constructed. 

The names should convey, by their mode of construction, 
as much information as possible ; they should have 
the greatest amount of independent significance 
which the case admits of. 

There are two ways of giving a name this sort of 
significance : — 






(1.) By making the name indicate, by its mode of forma- 
tion, the very properties it is designed to convey ; 
such as are sure marks (as chemical composition is) 
of all other properties. Chemical names are ex- 
amples, as "protoxide of iron." This, however, is 
seldom practicable. 

(2.) By making the name express the natural affinities of 
the group. This is accomplished by incorporating 
the proximate generic name with the specific, as 
"Felis leo." Even a ternary nomenclature, by in- 
corporating the next higher generic name, has been 
used, as " Khombohedral Lime llaloide." 



CLASSIFICATION, ETC. 229 

IV. WheweH's theory that Natural Groups 
are constituted by type, not by Definition. 

(The meaning of this is simply that objects are aggregated into 
Natural Groups on the basis of mere general resemblance 
(see p. 32), that is, what "Whewell calls by reference to a 
type, and not by resemblance in specific assignable par- 
ticulars which can be expressed in a definition.) 

A "type" is an eminent example of any class, i.e., an 
example which presents the characteristics of the 
class most conspicuously and completely. Natural 
classes, according to Whewell, are formed by being 
gathered round these types ; and a class really con- 
sists of the type, and all objects which bear a certain 
amount of general resemblance to the type. 

Mill's criticism : — 

Natural groups are determined by characters {i.e., 
by Definition, which enumerates those characters), 
not by Type or mere general Eesemblance ; but there 
is this amount of truth in Whewell's view : — 

1. It is not, as already said, necessary for every member 

of a natural group to possess all the characters laid 
down as those of the group ; and so far the defini- 
tion may be said to fail in determining the group. 
In fact, natural classes might be defined in this way 
— those things which either possess such and such 
characters (those enumerated in the definition), or 
resemble those things which do possess them more 
nearly than they resemble anything else. 

2. Our general conception of the group is a type, to 

which we usually in the first instance refer as a 
ready means of suggesting to what group aoy given 



230 CLASSIFICATION BY SERIES. 

individual or species will most probably belong ; but 
a determination of the question must always rest 
upon a reference to the characters laid down in the 
definition of the group. Natural grouping may, 
then, be said to be suggested by type {i.e., by mere 
general resemblance), but determined by definition 
(i.e., by possessing specific characters or properties). 



CHAPTEE VIII 

Classification by Series. 

Inasmuch as Zoology presents the best example of Classifica- 
tion by Series, it may be taken as the special example, and 
the phenomenon " Animal life n as the phenomenon we are 
supposed to wish to study. 

I. The subject generally. 

The object of Classification into Natural Groups is, as 
already stated, to make us think of those objects 
together which have the greatest number of im- 
portant common properties, and which, therefore, 
we have oftenest occasion, in the course of our In- 
ductions generally, for taking into joint considera- 
tion. 

Lut when our object is to facilitate the inquiry into 
some particular phenomenon! more is required. The 
classification must then bring the objects together 
in .such a manner that the simultaneous contempla- 



CLASSIFICATION BY SERIES. 231 

tion of them will throw most light upon that par- 
ticular subject. We must arrange the various groups 
into a Series, following one another according to the 
degrees or perfection in which they severally exhibit 
the phenomenon. The phenomenon itself, there- 
fore, must form the guiding principle of such an 
arrangement. 
It is evident that this serial classification, according to 
the degrees of the phenomenon of which the laws 
are to be investigated, puts the instances into the 
order required by the Method of Concomitant "Vari- 
ations, which, as already pointed out, is often the 
only available method in the case of phenomenon 
which we have but limited means of artificially 
separating. 

II. The requisites of a classification of this 
kind are : — 

1. To bring into one grand class all kinds of things 

which exhibit the phenomenon, in whatever variety 
of form or degree. 

2. To arrange these kinds into a Series, according to the 

degree in which' they exhibit it ; beginning with 
those which exhibit it in the greatest intensity and 
perfection, and terminating with those which exhibit 
least of it. 

Thus the phenomenon being, as supposed, " Ani- 
mal life," the first step, after forming a distinct 
conception of the phenomenon itself, is to erect 
into one great class — that of "animals" — all the 
known kinds of objects in which that phenomenon 
presents itself. "We must, in the next place, arrange 
the various kinds in a series, those which exhibit 



232 CLASSIFICATION BY SERIES. 

animal phenomena in the highest degree, as man, at 
the top, and gradually decreasing as we go down. 

It may happen that the arrangement required for 
the special purpose coincides with that required for 
general purposes ; this will naturally happen when 
the special phenomenon we are studying is so im- 
portant as to determine the main of the properties 
generally. 

III. The assumption of a Type Species is 
indispensable in inquiries of this kind. 

By a Type Species is meant that one amongst the differ- 
ent kinds which exhibits the property we are study- 
ing in the highest and most characteristic degree. 

This assumption of a type is necessary, because : — 

1. We must study the phenomenon in its highest mani- 

festations, in order to qualify ourselves for tracing 
it through its less obvious forms — for recognising 
the identity of the phenomenon under all its varia- 
tions. 

2. Every phenomenon is best studied, ecet. par., where it 

exists in the greatest intensity ; it is then that 
effects, which depend either upon it or upon the 
same cause with it, will exist in the greatest 
degree. 

3. The phenomenon, in its higher degrees, may be at- 

tended by effects or collateral circumstances which, 
in the smaller degrees, do not occur at all. 

IV. How the internal distribution of a series 
may most properly take place, — in what man- 



CLASSIFICATION BY SERIES. 233 

ner it should be divided into orders, families, 
and genera. 

1. The main principle of division, of course, must be 

natural ; the classes formed must be natural groups. 

2. But the principle of natural grouping must be applied 

in subordination to the principle of a natural series, 
this series having its arrangement determined by 
the variations in the particular phenomenon ; break- 
ing it into primary divisions, if possible, at the 
exact points where variations in the intensity of that 
phenomenon begins to be attended with conspicuous 
change in the general properties of the objects. 

3. In like manner each primary division should be so 

subdivided that any one portion shall stand higher 
than the next below in respect of the special pro- 
perty, or set of properties, we are studying. 

V. Finally, though the kingdoms of organ- 
ised nature afford, as yet, the only complete 
example of scientific classification, and the 
animal kingdom the only complete example of 
classification by series, yet the same principles 
are applicable in all cases where mankind are 
called upon to bring the various parts of any 
extensive subject into mental co-ordination. 
The proper arrangement, for instance, of a 
code of laws must depend upon similar con- 
ditions. 



BOOK V 



FALLACIES, 



I. Fallacies in general. 

A Fallacy is an argument in which inconclusive or ap- 
parent evidence is made the basis of a belief ; and a 
catalogue of the varieties of apparent evidence {i.e., 
evidence which, while seeming to be real and con- 
clusive, is not so) is an enumeration of fallacies. 

II. We do not include amongst Fcdlacies — 

1. Mere blunders — errors arising from a casual 
lapse, like a mistake in working an arithmetical 
sum while the general mode of procedure is correct. 

2. Moral sources of error, which are : — 

(a.) Indifference. 

(/;.) Bias — the most common being bias by our wishes, 

but very frequently also by our fears. 
The moral causes of error in reasoning, though most 



FALLACIES. 235 

powerful, are yet indirect ; they are but remote 
causes, and can only act through, the intellectual 
causes ; and to guard against these last is to guard 
against every other source of error. 

III. Classification of Fallacies (see first page 
of table). 

The five great classes into which Fallacies are 
divided are : — 

1. Fallacies of Simple Inspection. 

2. „ Observation. 

3. „ Generalisation. 

4. „ Eatiocination (Syllogistic). 

5. „ Confusion. 

The propositions which are not evidence of a particular 
conclusion are of course innumerable, and no classi- 
fication can be based upon that merely negative 
property ; but we may base it upon the positive 
property of appearing to be evidence; and we may 
arrange fallacies either (1.) according to what makes 
the evidence appear to be evidence, not being so (as 
the fact of its not being distinctly understood), or 
(2.) according to the particular kind of evidence it 
simulates (Inductive or Deductive). Mr Mill's 
classification is based on these principles jointly. 

As it is seldom that insufficient evidence, when clearly 
understood and unambiguously expressed, would 
not be seen to be fallacious, more or less of the 
element of Confusion enters into most fallacies ; 
but the class "Fallacies of Confusion" is reserved 



236 FALLACIES. 

for those in which Confusion is the chief, if not the 
sole, cause of the error. 

Mr Mill's classification may be briefly sketched 
out thus : — 

First, where the conclusion is assumed without there 
being any evidence to support it, — where it is be- 
lieved as a " self-evident axiom," — " Fallacies of 
Simple Inspection," "a priori Fallacies," or " Natural 
Prejudices ; " and Second, where there is some evi- 
dence — " Fallacies of Inference." This last is sub- 
divided according as (1.) the evidence is not dis- 
tinctly understood (i.e., not clearly seen to be what 
it really is), which gives us the " Fallacies of Con- 
fusion ;" and (2.) as the evidence is distinctly under- 
stood. This last is again subdivided according as 
the evidence consists of (a.) particular facts (Induc- 
tive), or (b.) general propositions (Deductive) ; and 
each of these is again subdivided according as (1.) 
the evidence is false, or (2.) is true, but inconclu- 
sive. 

It must not be supposed that any given fallacy can 
always be referred absolutely to one or other of the 
great classes. Except Fallacies of Confusion hardly 
any fallacy can be assigned to its proper place till it 
has been expressed at full length ; and the mode of 
doing this is often a matter of choice. All that we 
can generally say, then, in any particular case, is, 
that if the intermediate steps in the argument be 
filled up in such and such a way, the fallacy will fall 
into such a class. 



FALLACIES. 




238 FALLACIES. 



I.— Fallacies of Simple Inspection. 

The following are examples of some principal 
forms : — 

(a.) That the Inconceivable is False. 

(b.) That everything which can be conceived in the mind 
must have a corresponding real existence in fact. 
(Realism an exaggerated form of this Fallacy.) 

(c.) The doctrine of the " Sufficient Reason," that a thing 
must be so and so, because we know of no reason 
why it should be otherwise. 

(d.) That the distinctions in nature must correspond to 
distinctions in language. (Common error with 
Greek philosophers.) 

(e.) That a phenomenon can have but one cause. (An 
error which vitiated Bacon's Principles of Inductive 
Inquiry.) 

(/.) That there must be a resemblance between a pheno- 
menon and its conditions. 



FALLACIES. 239 



II.— Fallacies of Observation. 

Here the error lies in overlooking or in mistaking 
something (i.e., in collecting our data), and therefore 
we have either : — 



((a.) Fallacies of ' Non-Obser- C Of Instances. 
vation. J Of Essential 

(overlooking) ( Circumstances. 

(6.) Fallacies of Mai- Observation — mistaking 
(seeing wrong) inference for percep- 
tion — believing that 
we have an imme- 
diate knowledge of 
something which we 
really infer. 



Fallacies 

of 

Observation. 



Non-Observation or neglect of instances may occur either 
(a.) From the circumstance that some of the instances 
are naturally more impressive than others, — as, for in- 
stance, positive against negative instances. We are 
very apt to notice instances in which a phenomenon 
occurs, without regarding the equally important in- 
stances in which it does not occur, (b.) From pre- 
conceived opinion, — the most fertile source of error 
of this kind. That which in all ages has made men 
unobservant of the plainest facts, is their being con- 
tradictory to first appearances or any received belief. 
Thus, for centuries it was universally held that a 
body, ten times as heavy as another, fell to the ground 
ten times as fast ; that the magnet exerted an irre- 
sistible force, and so on. 



240 FALLACIES. 

III.— Fallacies of Generalisation. 

Here we have rightly obtained the obtainable 
evidence bearing on the conclusion ; but we have 
wrougly concluded from it. The error lies in making 
the Inference, not in collecting the data. 

This class of Fallacies — the error of drawing con- 
clusions from insufficient evidence — is the most ex- 
tensive of all, as might indeed be anticipated. It is 
only possible, therefore, to indicate some of the prin- 
cipal sub-classes : — 

- (a.) Generalisations which cannot in the nature of the case 
be established, where we have no real data or 
evidence to argue from, — as, for example, inferences 
as to what may go on in remote parts of the universe. 
(b.) All propositions which assert impossibility (universal 
negative propositions), except those which assert 
mathematical truths or the impossibility of excep- 
tions to the universal law of causation, 
(c.) All generalisations which profess to resolve radically 

different phenomena into the same. 
(d.) The fallacy involved in placing mere empirical laws 
(and those often of the lowest kind) on the same 
footing of generality as true casual laws. As — 
(1.) Empirical laws generalised from mere nega- 
tions. (" What has not happened, never will") 
(2.) Empirical laws arrived at merely by the 
" induction by simple enumeration." 
(«.) Generalisations which improperly infer causation. 

[Fallacia non caiisa pro causd.) 
(/.) Arguments from false analogy. {Fallacia non talis 
K. pro tali.) 

Avoid confusion between "Ind. by Simple enumeration;" and "Ind. by 
Complete enumeration;" In the format ire conclude that a law is true simply 
H e have never met With an instance to the contrary ; the latter is the 
same m the "Mere Verbal Transformations" of Mill. 






fe 



FALLACIES. 241 



IV.— Fallacies of Ratiocination. 



Fallacies of 
Katiocination. 



1. Fallacies of Immediate Inference (as in 

the Conversion, Opposition, iEquipol- 
lency of Propositions). 

2. Syllogistic Fallacies ( = the "Logical 

Fallacies"'' of Whately). Undistri- 
buted middle, illicit process of major 
and minor, and so on. 

3. " Changing the ( Secundum quid. 

Premisses " \ Per accidens. 



The meaning of the phrase, " Changing the Premisses," 
applied by Mill to a certain class of the Fallacies of 
Katiocination, is this : — A premiss in a syllogism is 
regarded as being the conclusion of some previous act 
of inference. Now, if the proposition, as laid down 
for a premiss, is really distinct from that which was 
proved, an error may easily arise in making a deduc- 
tion from it, — a change is made in passing from the 
proposition as a conclusion to the proposition as a 
premiss. The Fallacia a dicto secundum quid ad dic- 
tum simpliciter (briefly designated " secundum quid"), 
and the Fallacia accidentis are important forms of 
this sub-class of Fallacies ; in both cases the error 
lies in laying down a major premiss too absolutely or 
generally, — more generally, in fact, than the evidence 
which supports it will warrant. Thus, if we say, "All 
men have a right to their personal liberty" it is clear 
that, generally speaking, we should really mean to 
limit it in some such way as this, — " All men, who 
are of sound mind, and who are not guilty of criminal 
conduct, have a right to their personal liberty." The 
evidence for the proposition only proves this more 
limited form ; and if we use it as a premiss without 
these tacit limitations, we may be guilty of a fallacy. 

Q 



242 



FALLACIES. 



V.— Fallacies of Confusion. 

Here the mistake lies, not so much in over-estimat- 
ing the probative force of known evidence, as in the 
absence of a distinct and definite conception of what 
that evidence really is, or what conclusion is required 
to be proved. 



Fallacies of 
Confusion 
are : — 



1. Ambiguous lan- 
guage (the "semi- 
logical" of 
Whately). 



2. Petitio Principii. 



3. Arguing in a 
circle. 



4. Ignoratio 
Elenchi. 



I F. equivocationis. 
F. amphibolice. 
F. figurce dictionis. 
F. compositionis. 
F. divisionis. 
F. plurium interroga- 
tionum. 

The employment of a pro- 
position to prove that 
upon which it is itself 
really dependent for 
proof. 

Proving two propositions 
reciprocally from one 
another, or more than 
two in a reciprocal man- 
ner. 

Proving part of a conclu- 
sion. 

Proving a conclusion vague 
from the use of complex 
and general terms. 

Fallacy of shifting ground. 

Fallacy of objections. 

Fallacy of special appeals, 
as " ad hominem" &c. 



FALLACIES. 243 

The Fallacy, "Ignoratio elenchi" — ignoring the elenchus 
— is the proving of a proposition resembling more or 
less the conclusion required, but not really identical 
with it, — a very common form of Fallacy. The 
elenchus being the contradictory of the assertion 
of the supposed opponent. 



I.— Fallacies of Simple Inspection. 

Here a proposition is either admitted as 
true upon a " simple inspection " of it, as a 
self-evident truth, without any extraneous 
evidence, or, perhaps, more commonly the case 
is that d priori considerations only create a 
presumption in favour of a proposition, so that 
it is accepted, not absolutely without evidence, 
but upon evidence which would be seen to be 
insufficient if the presumption did not exist. 

Amongst the many forms in which such errors 
may be presented, are : — 

1. That the reality of a thing ivill follow the idea 
of it ; that the idea is either a prognostic, or 
even a cause of the thing thought of. 

This is extensively exemplified in many popular supersti- 
tions : the Komans, for instance, would not mention 
unlucky words, as " death." 



244 FALLACIES. 

2. That a wonderful or precious tiling must have 

wonderful properties. 

Gold regarded as the universal medicine. 

3. Things we cannot help thinking of together 

must coexist. 

Thus, it is often argued that B must accompany A in 
fact, because B is involved in the idea of A. This 
argument is at most an appeal to the authority of 
our predecessors. The doctrine that whatever the 
idea contains must have its equivalent in the thing, 
pervades the philosophy of Descartes, Leibnitz and 
Spinoza, and the modern German metaphysicians. 

4. The inconceivable is false. 
This has been already examined. 

5. That everything ivhich can be conceived in the 

mind must have a real existence in fact. 

Realism was an exaggerated form of this fallacy, — argu- 
ing, because a general idea of " man " can exist in 
the mind, there must be something really existing 
corresponding to that idea, just as when we think 
of any particular man there is a real corresponding 
existence. 

6. The principle of the sufficient Reason. 

That is, a phenomenon must follow a certain law, be- 
cause we can see no reason for its deviating from 
it in one direction rather than another. Thus, 
Mathematicians argue that a body at rest cannot 



FALLACIES. 245 

begin to move, because if it did, it must move in 
some particular direction, and we can see no reason 
why it should move in one direction rather than 
another.-, it will not move at all. But the fact of 
our being able to see no reason, is not always a proof 
that no such reason exists. 

7. That the differences in Nature correspond to 

our received distinctions in names and classi- 
fications. 

This fallacy prevailed to an extraordinary extent amongst 
the Greek philosophers, who imagined that by an 
analysis of words they could discover facts. 

8. That a phenomenon can only have one cause. 
This was the error which misled Bacon. 

9. That the cause or conditions of a phenomenon 

must resemble that phenomenon. 

This does sometimes happen, — motion may produce 
motion, — but very commonly no resemblance what- 
ever can be traced between an effect and its cause. 



II. — Fallacies of Observation. 

The term " observation " is here equivalent 
to the ascertainment of the facts upon which an 
Induction is grounded, however obtained, — 



i 



246 FALLACIES. 

whether by direct experience or by inference 
from something else. 

A fallacy of observation, then, may be either negative or 
positive; negative or non- observation when all the 
error consists in overlooking something which 
might have been known, and which, if known, 
would make a difference in our conclusion ; posi- 
tive, mal- observation, when something is not simply 
unseen, but seen wrongly; when a fact or pheno- 
menon, instead of being taken for what it really is, 
is mistaken for something else. And, as we have 
previously observed, the senses cannot properly be 
said to be capable of error, but only inferences from 
sensations, this kind of fallacy can only happen 
when something which has really been erroneously 
inferred is supposed to have been actually observed. 

As regards non-observation, we may overlook either (1.) 
Instances, or (2.) Essential circumstances in those 
instances (see table). It maybe added that neglect 
of instances does not per se and necessarily vitiate 
the conclusion, unless we at the same time neglect 
to eliminate chance, which error would come under 
the next head, — Fallacies of Generalisation. 



III. — Fallacies of Generalisation. 

In addition to what is given in the table, 
we may notice the following points : — 

Generalisations which profess to resolve radically 
different phenomena into the same. 



FALLACIES. 247 

Whenever our consciousness recognises between two of 
its states a radical distinction; whenever we feel 
that no mere adding on of the one phenomenon to 
itself would produce the other (as it would if the 
difference were only in degree), the two states must 
be the result of the operation of radically different 
laws, and any attempt to resolve the one into the 
other must be futile. (See also Book III., chap, 
xiv.) 

Undue extension of Empirical Laws. 

As examples of the kinds of Empirical Laws which 
are often unduly extended, we have — 

(1.) Empirical Laws generalised from mere negations — 
their formula being " whatever has never happened, 
never will ;" as " negroes have never been so highly 
civilised as whites, therefore they never will," and 
such like. 

(2.) Empirical Laws, though based on positive data, yet 
only established by an Induction per enumerationem 
simplicem, stand one degree higher in the scale, but 
still ought only to be extended to adjacent cases. 

Generalisations ivhicli improperly infer causa- 
tion — 

The most common is the fallacy known as "the post 
hoc ergo propter hoc" arguing that B is caused by 
A, because B follows A. 

Arguments based on false analogy — 

For the conditions which determine the probative force 
of an analogical argument, see Book III., chap. xx. 



248 FALLACIES. 

The most fertile source of fallacies of generalisation 
is bad classification, — bringing together under 
a common name things which have no common 
properties, or at least no peculiar common pro- 
perties. 



IV. — Fallacies of Katiocination (see table) . 

[By Immediate Inference is meant the 
direct deduction of one Proposition from 
another or others, without the intervention of 
a middle term. The following are some of 
the common forms of this kind of Inference 
(they would come under Mill's class of " Infer- 
ences improperly so called "). 

1. Immediate Inferences by conversion — 

All men are mortal ; 
■"■ Some mortal beings are men. 

2. By opposition — 

It is true that all men are mortal ; 
■\ It is false that some men are not mortal. 

3. By added determinants — 

A negro is a fellow-creature : 
.\ To murder a negro is to murder a fellow- 
creature. 



FALLACIES. 249 

4. By fusion of judgments — 

A negro is a fellow-creature ; 
Honesty deserves reward ; 
■■. A negro who is honest is a fellow-creature deserving 
of reward. 

It is evident that a fallacy may lurk in processes of 
this kind.] 

Syllogistic fallacies include all which offend against 
the laws of Syllogism. 



V. — Fallacies of Confusion. 



Fallacies of ambiguous language — 

Here the premisses are verbally sufficient to prove the 
conclusion, but not really ; they are the same as 
the " semi-logical " of Whately. 

Petitio Principii — 

The employment of a proposition to prove that upon 
which it is itself really dependent for proof, by no 
means implies that degree of inattention or im- 
becility which might seem at first sight involved 
in such an error. We must remember that even 
philosophers hold many opinions without exactly 
remembering how they came by them, or upon what 
evidence they were based; and in such a case they 
may easily be betrayed into deducing from them the 
very propositions which are alone capable of serv- 
ing as premisses for their proof. 



250 FALLACIES. 

Arguing in a circle — 

Is an attempt to prove two propositions reciprocally 
from one another, or three or more propositions in 
a similar manner. Thus, A is true because B is, 
B is because C is, C is because A is. This form of 
error is, however, more frequent in the form of 
simply admitting two propositions which can only 
be proved from one another, than as a deliberate 
attempt to do this. 

Of course, a proposition would not be admitted merely 
as a corollary from itself, unless it were so expressed 
as to seem different \ this is often done by stating 
one proposition in the concrete, the other in the 
abstract form, or one in Saxon, and the other in 
Classical phraseology. 

Ignoratio elenchi — 

Ignoring the elenchus ("the elenchus" being the con- 
tradictory of the assertion of the supposed oppo- 
nent), is the proving of a conclusion more or less 
like the one required, but not really identical with 
it. This is a very common form of fallacy. 



BOOK VI. 



LOGIC OF MORAL SCIENCES. 



CHAPTEKS I. II. and III. 

By the Moral Sciences we mean those relating to the human 
mind and to human society ; these form the most complex 
problems which can be submitted for our consideration, and 
it remains in this book to determine the method of scientific 
inquiry most likely to lead to satisfactory results in con- 
nexion with these questions. But, first^it is necessary to 
obviate an objection that may be made to the effect that 
human actions are not the subject of law, and, therefore, not 
of Science. 

Liberty and Necessity. 

The great problem of " the freedom of the will " has been 
much obscured by the inappropriate use of the word 
" necessity," — which, in these discussions, must be 
understood as equivalent to certainty, and not to 
compulsion. 

The doctrine of philosophical necessity, or of the uniformity 



252 LOGIC OF MORAL SCIENCES. 

of the sequence between motives and actions, is 
simply this — given the motives present to the mind 
of an individual, and given also his character and 
disposition, the manner in which he will act may 
be unerringly inferred. This is proved by the uni- 
' versal experience of mankind ; whenever we rely 
upon a human being acting in a particular manner, 
we rely upon the uniformity of the sequence be- 
tween motives and actions ; and a most convincing 
proof is presented by statistics, which show the 
uniformity of the occurrence of apparently casual 
acts, when we observe on a scale sufficiently large 
to eliminate chance. It is sometimes said that 
our consciousness proves to us that the will is free, 
meaning by this that its acts are spontaneous, un- 
caused. But consciousness testifies nothing of the 
kind, it only testifies that we are under no compul- 
sion ; but the law does not assert this, — it simply 
asserts that the act follows the motive causes by a 
certain and unconditional sequence, it is no more a 
question of constraint than in the case of physical 
causes and effects. 
There may be, therefore, a Science of human nature. 
Such a Science cannot, however, be a Science of 
exact predictions, but only of tendencies, since the 
causes are too uncertain to enable us to go beyond 
this. 



CHAPTERS IV. and V. 
Laws of Mind. 

The laws of mind are the laws by which one state of 
mind is produced by another. The simple laws of 



LOGIC OF MORAL SCIENCES. 253 

mind must be ascertained by experiment; the com- 
plex laws are results of these, either by way of 
composition of causes, or as Heteropathic Effects. 
The mental differences between individuals are 
generally not ultimate facts, but are the results of 
differences in the mental history, education, cir- 
cumstances, &c. 
Although mankind have not one universal character, 
yet there are universal laws of the formation of the 
character (laws of Ethology). These laws cannot 
be discovered experimentally ; the Deductive 
Method is the grand agent, observation being only 
valuable as affording the means of verifying its 
conclusion ; the object of the Science of Ethology 
being to determine from the general laws of mind, 
combined with external circumstances, the condi- 
tions which aid or check the growth of good or bad 
qualities. Education will then consist in applying 
these results. 



CHAPTEES VI VII. VIII. and IX. 

The Social Science. 

Social phenomena, being the phenomena of human 
nature in masses, must obey fixed laws, since human 
nature is subject to the same. The Social Science 
can never be a science of positive predictions, but 
only of tendencies, like most of the propositions 
relating to human nature. There have been many 
attempts to investigate this science, and to build 
up systems. A consideration of two methods, 



254 LOGIC OF MORAL SCIENCES. 

erroneously used, the Experimental and the Ab- 
stract Deductive, may with advantage form a pre- 
face to the true method. 

The Experimental or Chemical Method in 
the Social Science. 

The followers of this method refuse to accept conclu- 
sions except they be based upon specific experience 
in all cases. This attempt must fail, for we have 
already pointed out that in complex effects, direct 
Induction is scarcely ever applicable. Here, on 
account of the number and complexity of the causes, 
this is pre-eminently true. 

The Abstract Deductive or Geometrical 
Method. 

Those philosophers who have applied this method in the 
treatment of questions of Socfial Science have been 
correct in so far as they have been aware that the 
method of that science must be Deductive, but have 
erred in taking the application of that method to 
sciences not concerned with causation (as Geometry) 
as the type of the method required here. Their 
usual plan has been to take some proposition or 
propositions as premisses or axioms, and from them 
to deduce and build up a system. This method 
was adopted by Ilobbes and by Blatham ; but it is 
not the true method. The Social Science is a science 
of causes, and causes may be counteracted, and 
hence its method must be that form of the Deduc- 
tive Method which is applicable to such sciences, 
namely, — 



LOGIC OF MORAL SCIENCES. 255 

The Concrete or Physical Deductive 
Method. 

That is to say, we must compound with one another the 
laws of all the causes on which any effect depends, 
and infer its law from them all. It is true we must 
often invert the order of our proceedings, and first 
obtain our conclusions conjecturally from specific 
experience, and then verify them by a priori 
reasonings. 

Sociology, we have already remarked, is a science, not of 
positive predictions, but only of tendencies ; and not 
only so, but its assertions must be hypothetical, and 
state the operation of a given cause in given circum- 
stances. It also answers best to divide the science 
into subordinate sciences, — each of which considers 
one great social cause. Thus, Political Economy 
considers society as influenced by the desire of 
wealth. 



CHAPTERS X. and XI 

Inverse Deductive or Historical Method. 

By a " State of Society' 7 is meant the simultaneous state 
of all the greater social facts or phenomena. Such 
are the degree of knowledge, of intellectual and 
moral culture, wealth, industry, social classes, laws, 
&c. 

Now, amongst these various phenomena there are cer- 
tain uniformities of coexistence ; that is to say, it is 



256 LOGIC OF MORAL SCIENCES. 

not any combination of these social facts which can 
coexist, but only certain combinations. Just as the 
various parts and states of the individual body have 
a constant reciprocal influence over one another, so 
it is in the body politic ; there is a consensus between 
the different social facts ; and the study of these 
uniformities of coexistence constitutes the science 
of Social Statics. 

But besides presenting phenomena of this kind, society 
is in a constant state of progress ; the state of 
society at any given time differs from its state at 
some previous time ; the study of the laws by which 
any state of society produces the state of society 
which succeeds and takes its place, constitutes 
Social Dynamics — the theory of society as pro- 
gressive. 

The evidence of history goes to prove that one great 
element is predominant over all others as the prime 
agent in determining social progress, — that is, the 
state of the speculative faculties, including the nature 
of the belief which men hold at the time, and the 
means by which they have arrived at them. And 
M. Comte has laid down one generalisation which 
he regards as the fundamental law of the progress 
of human knowledge, — viz., that speculation on 
every subject has three successive stages, — first, 
when the tendency is to explain phenomena by 
supernatural agencies ; second, by metaphysical 
abstractions (as nature, vital force, &c.) ; and third, 
when it confines itself simply to ascertaining their 
laws of succession and coexistence. This is an 
example illustrative of the great doctrine which is 
laid down in this chapter, that the collective series of 
social phenomena, the course of history, are subject to 
general laws which philosophy may possibly detect. 



LOGIC OF MORAL SCIENCES. 257 



CHAPTEK XII. 

Logic of Practice or Art 

In speaking of the logical method of "Art" that term is 
used in the sense of a body of rules directed to some 
practical end, as when we speak of the "Art of 
Building," " of Government," and so on ; and not as 
having reference to the poetical or aesthetic aspect 
of things. 

Art, then, is characterised by expressing its propositions 
in the imperative mood ; it speaks in rules or pre- 
cepts, as contrasted with the direct indicative asser- 
tions of science. 

The logical method of art may be summed up thus : — 
Every art starts from a single major-premiss — that 
such and such an end is desirable. Science, then, 
investigates the means by which the end can be 
secured ; and this being accomplished, it hands over 
the necessary propositions to Art to be turned into 
practical rules. 

In order to know what things are really desirable, 
we require a Science of Teleology, or of ends (i.e., 
things desirable) ; but however many subordinate 
ends may be allowed, there must be one single 
ultimate end to which all others may, in the last 
resort, be referred. This final standard is the pro- 
motion of the happiness of all sentient beings. 

Ethics or Morality is an Art, corresponding to a portion 
of the Sciences of Human Nature and Society ; its 
method must, therefore, be that of Art in general. 

R 



APPENDIX. 



Connotative Names. 

By the " Connotation" of a name is understood the attri- 
butes which we mean to assert that an object pos- 
sesses, when we predicate the name of that object. 
Thus, if we assert that an object before us is a 
"man," we mean to convey that that object pos- 
sesses certain attributes — animality, rationality, and 
two-handed upright form. These attributes con- 
stitute the connotation or meaning of the name 
" man." 

The mode in which some Logicians have repre- 
sented the point is by speaking of the idea as "com- 
prehending " or including other ideas ; the idea of 
"man" would be said to comprehend the idea of 
animality^ rationality, &c. What the idea compre- 
hends is, in fact, precisely equivalent to what the 
name connotes ; and the definition is spoken of as 
being the unfolding or stating in words either of the 
comprehension of the idea or of the connotation of 
the name. Connotative names are sometimes 
spoken of as being " significant marks." 



260 APPENDIX. 

Non-connotative Names 

Are simply marks and nothing more — "non-significant 
marks." If every house in a town has its own 
letter or number of some kind on the door, such a 
number or letter would be a mark of the corre- 
sponding house, but it would signify nothing, con- 
vey no meaning. Such exactly are non-connotative 
names, — the chief of which are proper names, and 
the names of simple attributes. " Caesar" is like a 
cross put on an individual, chiefly to identify him, 
and save the trouble of a long description ; but it 
conveys no meaning. If we are told that an object 
is called "Caesar," we should know from that 
nothing of its properties or attributes ; it might be 
a man, dog, horse, &c. But if instead of speaking 
of an object as "Caesar," we speak of it as a 
" Roman general," this is not only a name, but a 
name significant of something, — viz., belonging to 
Rome and being a general, and therefore is conno- 
tative. 

The following classes include the most important 
Connotative Names : — 

1. All concrete general names, — as "man," animal," 

" planet," — the names of classes of objects. Such a 
name evidently connotes the attributes, the posses- 
sion of which makes any object a member of the 
corresponding class. 

2. Descriptive individual names — that is, names which, 

instead of designating an individual by a mere un- 
meaning proper name, point him out by some quali- 
ties or properties or marks which belong to him. 
Thus the name "Gladstone" is a mere mark, and 



APPENDIX. 261 

therefore a non-connotative name, while "present 
Prime Minister of England " refers to the same in- 
dividual, but is connotative, i.e., it implies attri- 
butes or properties. The former name conveys in 
itself no information ; anything which we happen to 
know of the person when the name is pronounced 
is merely accidental ; but the latter does tell some- 
thing, wholly independent of such casual know- 
ledge, and equally to every person, however well or 
ill informed with respect to the individual in ques- 
tion. Some authors on Logic (see " Shedden's 
Manual," p. 17, &c.) have maintained that proper 
names are connotative ; by this they mean that, for 
example, " Gladstone " connotes " a politician, in 
1869 Prime Minister of England," &c, because such 
circumstances may be brought to mind by the 
mention of that name ; so that they hold that 
proper names connote whatever any given indi- 
vidual in whose hearing they are pronounced may 
happen to know of the person to whom the name 
belongs. The name "John," for instance, being 
universally and exclusively applied to males, would, 
according to this view, connote to every one the 
attribute "masculinity," — the being of the male 
sex ; while to any individual who happened to know 
the particular John referred to, it would connote 
anything whatever that he might happen to know 
or remember about him. Now, to take the stronger 
of these cases, — that is, where the circumstance 
associated with the name is of such a character that 
the mention of the name would suggest the circum- 
stance to every one, as the name "John" would 
that the person spoken of was a male. The name 
John, then, we are to suppose, connotes " being of 
the male sex ; " but every name whose (entire) con- 



262 APPENDIX. 

notation consists of a given attribute or set of attri- 
butes, is the name of a class of objects, — viz., those 
objects which possess the connoted attributes, — and 
to every one of these objects the name is applicable.* 
The name John, however, is not the name of every 
individual of the male sex, and this consideration 
shows conclusively that the attribute " masculinity " 
does not constitute the connotation of that name. 

The distinction, in fact, is obvious enough be- 
tween what a name really means, and what we may 
happen to know in some way or other about the 
object to which it is applied. Of course the term 
"connotation" may have its meaning extended to 
include such casual associations, but if we assign 
such a meaning to it, we must remember that it is 
then totally distinct from the connotation which Mr 
Mill so constantly refers to. Accidental knowledge 
of the sort we are discussing is of no importance in 
Logic, and to incorporate it with true connotation 
would destroy all the value of the distinction, and 
constitute a mischievous distortion of the recognised 
signification of the word. 
3. Certain Abstract Names, — or names of attributes, 
viz., those abstract names which are names either — 
(1.) Of attributes which have attributes ; or (2.) Of 
groups or aggregates of attributes. Thus, " civilisa- 
tion " is the name of an attribute (the correspond- 
ing concrete being " civilised beings "), which 
includes a number of other attributes, — a group, 
in fact, bound together by the name, — such as in- 

• We say "entire" for this reason,— suppose a name to connote 
attributes A, B, and C, and nothing more, then to every object which 
possesses the three attributes A, B, and C, the name is applicable, 
bat not necessarily to objects which possess only one or two of these, 
as is indeed self-evident. 



APPENDIX. 263 

tellectuality, moral and aesthetic cultivation, and 
so forth, and these attributes form the connotation 
or meaning of the name "civilisation." As an 
example of an attribute which possesses attributes, 
Mr Mill gives " faultiness," — the name of some 
quality which has the attribute " causing incon- 
venience," which, therefore, the abstract name 
" faultiness " connotes. 

Non-connotative Names require but little illustration, — 
they are simple marks without meaning. The chief 
classes are — (1.) Proper names ; and (2.) The names 
of simple, unanalysable attributes, or, in other 
words, of our elementary feelings. " Whiteness," 
for example, is a mark put on a certain quality of 
objects, just as " Csesar " is a mark put on a certain 
individual, 

By the Denotation of any name is understood the whole 
collection or aggregate of objects to which the 
name is applicable. The denotation of " man " 
includes every human being ; of " law," every law ; 
of " crime," every criminal act ; and so in every 
case. Many logicians use the term " extension " of 
a name as equivalent to its denotation. 



264 APPENDIX. 



II. 



Process of forming general notions {con- 
cepts, general ideas). 

Take the name "man," — in connexion with this we have 
three things, — the name itself, the class to which it 
corresponds (i.e., its denotation), and the general 
notion or idea which is raised up within our minds 
by the mention of the name. 

What is the nature of the process by which any 
such general notion is obtained, or by which such a 
class is formed ? Let us place ourselves in the posi- 
tion of the first intelligent observer of nature ; he 
would be continually encountering a variety of ob- 
jects, and after a little experience he could not help 
noticing that certain of these resemble each other, 
the resemblance consisting in the possession of 
certain common attributes. Thus he would meet 
with objects which we now recognise as forming the 
class " liquids ;" he would, on instinctively com- 
paring such together, find that they agreed in pos- 
sessing certain properties (perfect molecular mobi- 
lity and inelasticity) ; and whenever he met with a 
fresh example, he would recognise in it these com- 
mon properties. In this way a general idea or con- 
cept of liquids would be formed, made up of these 
common qualities ; a class would be formed con- 



APPENDIX. 265 

sisting of all objects which possessed them, and a 
general name might be imposed on that class, which 
name would connote or imply those same properties, 
— that is, the applicability of the name to any given 
object would depend upon that object possessing 
them. 

It is evident, moreover, that the name, once given, 
serves ever afterwards to bind and keep together 
that group of attributes ; were it not for the name 
we should be almost sure, sooner or later, to forget 
our classification, and have to make it over again, 
and even if not, we could not permanently register 
its results to communicate to others, or transmit to 
our successors. 



The process, then, of forming general notions may 
be summarised thus : — 

1. The senses, and memory reproducing their impressions, 

are continually giving us a knowledge of a succes- 
sion of different objects. 

2. Comparison of certain of these objects shows that 

they are similar, and we recognise the similarity as 
consisting in the possession of certain common 
attributes. 

3. Our attention being thus concentrated upon these 

common qualities, the mind instinctively binds 
them up into an aggregate or group, which forms 
our idea of the class, that is, a general notion. 

The associative law of " Similarity " (see Bain's " Senses 
and Intellect," 2d edition, pp. 525-533, on entire 
subject), is the principal intellectual faculty employed 
in this process. 



266 APPENDIX. 

Different views as to the form in which 
general notions exist in the mind. 

A general notion may be denned, as we have seen, as a 
conception of a multitude of individuals as an 
assemblage or class. Such general notions certainly 
exist in the mind in some form ; we can, somehow, 
conceive of a multitude of individual objects as an 
assemblage or class, or we could not use general 
names with any consciousness of meaning. 

Before giving a summary of the different views which 
have been held on this question, we may remark, 
that it is not strictly any part of Logic ; it is suffi- 
cient for its purposes that the name of a class call 
up some idea by which we can, to all intents and 
purposes, think of the class as such. Mr Mill 
however, gives the following different doctrines on 
the question. As to the nature of the idea called 
up by a general name : — 

(a.) Doctrine of Locke, Brown, and the conceptualists, 
— that a general idea is composed of the various 
circumstances in which all the individuals denoted 
by the general name agree, and of no others. 

(b.) Doctrine of James Mill, — that such an idea is that 
of a miscellaneous assemblage of the individuals 
belonging to the class. Thus, the name " man " is 
supposed to call up the idea of an assemblage or 
mass of human beings. 

(c.) Doctrine of Berkoly, Dugald Stewart, and the 
modern Nominalists, — that the idea of a class is 
really the idea of some one individual of that class 
with his individual peculiarities, but with the ac- 
companying knowledge that such peculiarities are 



APPENDIX. 267 

not found in every member of the class, — that is 
are not properties of it. 

(d.) Bailey's view is, that the general name raises up an 
image, sometimes of one known individual of the 
class, sometimes of another ; not unfrequently of 
several such individuals in succession, and some- 
times an image made up of elements from different 
objects. 

(e.) In a very large number of cases, where a general 
name is mentioned, no distinct idea whatever is 
called up in the mind ; the name is used as a mere 
symbol, employed as an x or an a in an algebraical 
process. 

It is impossible to discuss the subject fully here ; it is 
only necessary to say, that the ideas called up by 
general names are certainly not always of the same 
kind. (Comp. " Symbolical and Notative Concep- 
tion " — Thomson's Outlines.) 



THE END. 



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